The “Last Supper” in Jn 13:2ff is the same meal/event described in Mt, Mk, & Lk.1 This is relevant because the meal/event described in Mt, Mk, and Lk is the Paschal meal.2
We know the Last Supper in Jn is the same as that in they synoptics for four reasons:
a. The meal/event occurs in the evening with all His apostles (including Judas).
b. During the meal/event, Jesus says “one of you will betray me.” The apostles then puzzle over who, followed by Jesus' bowl-dipping/morsel identification of Judas (Jn 13:21, Mk 14:18-21, Mt 26:21-25, Lk 22:21-23)
c. During the meal/event, Peter then proclaims unswerving loyalty; Jesus predicts Peter's 3 denials before the rooster crows. (Jn 13:37-38, Mk 14:30-31, Lk 22:33-34, Mt 26:31-35)
d. Directly after that meal/event, Jesus prays in the garden, where He is betrayed by Judas and arrested (Mt 26:36–46; Mk 14:32–42; Lk 22:39–46; Jn 18:1). Between then and the next evening, Jesus is put on trial and crucified.
The meal/event described in Mt, Mk, & Lk is the Paschal meal (initiating the 7-day “Feast of the Passover”):
Mk 14:12-16 -- On the first day of Unleavened Bread, when the Passover lamb was being sacrificed, His disciples said to Him, "Where do You want us to go and prepare for You to eat the Passover?” [...] 14and wherever he enters, say to the owner of the house, 'The Teacher says, "Where is My guest room in which I may eat the Passover with My disciples?"” […] The disciples went out and came to the city, and found it just as He had told them; and they prepared the Passover.
Mt 26:17-19 -- Now on the first day of Unleavened Bread the disciples came to Jesus and asked, “Where do You want us to prepare for You to eat the Passover?” 18And He said, "Go into the city to a certain man, and say to him, 'The Teacher says, “My time is near; I am to keep the Passover at your house with My disciples."'” 19The disciples did as Jesus had directed them; and they prepared the Passover.
Lk 22:7-8, 11 -- Then came the first day of Unleavened Bread on which the Passover lamb had to be sacrificed. 8And Jesus sent Peter and John, saying, “Go and prepare the Passover for us, so that we may eat it.” […] 11“And you shall say to the owner of the house, 'The Teacher says to you, “Where is the guest room in which I may eat the Passover with My disciples?'”
Jn 13:1 “before the Passover Feast" (a 7-day festival) sets Jesus' love right before the footwashing episode.1 This is relevant because Verse 1 set us “before” the Passover feast, so after the footwashing, the following meal most likely has in mind the very Passover just mentioned.2
[Setting the stage]: Jn 13:1 [NIV] -- It was just before the Passover Feast. Jesus knew that the time had come for him to leave this world and go to the Father. Having loved his own who were in the world, he now showed them the full extent of his love.
[The Feast starts, Jesus shows Love] 13:2-4 The evening meal was being served, and the devil had already prompted Judas Iscariot, son of Simon, to betray Jesus. Jesus knew [...] so he got up from the meal, took off his outer clothing, and wrapped a towel around his waist. After that, he poured water into a basin and began to wash his disciples' feet, ... ... ...
Donald Carson (NT Scholar & Prof.; Ph.d in Phil.): “[t]here is nothing in the words themselves to discourage us from taking the clause [v1 "just before the Passover feast"] as an introduction to the footwashing only, and not to the discourses that follow the meal. Chronologically, the opening words then place the footwashing before the Passover meal about to begin.” [Pillar NT Commentary: John (W.B. Eerdmans, 1991) 460.]
Craig Blomberg (Prof. of NT at Denver): “[a]ll v.1 says is that before the Passover Jesus knew what was about to happen and was prepared to love his followers to the end. Without any further qualification, the meal described in v.2 is most naturally understood as the Passover meal, in light of the reference to Passover in v.1 (Ridderbos, John, 455). When Jesus leaves in vv. 29-30 to buy provisions for the "feast," the rest of the week-long festival is in view (Carson, John, 475). The same is true of 18:28 (Kostenberger, John, 524.)” [Jesus and the Gospels: An Introduction and Survey, 2nd Ed. (B&H Publishing, 2009) 385fn.]
Jn 13:12 says they all “reclined at the table.”1 This is relevant because reclining is “a feature unusual in Jewish evening meals, but mandatory for the Passover celebration in early Judaism.” [George Murray, Word Biblical Commentary: John (Word, Inc., 2002), 225.]2
Barry Smith (Ass. Professor of Philosophy & Religious Studies): “Jesus and his disciples reclined at the table (John 13:23, 25), indicating that the meal was not an ordinary one. Rather, it was a festival meal, and, given the context, it could only have been a Passover meal.” [“The Chronology of the Last Supper,” Westminster Theological Journal 53:1 (1991): 32.]
Roger Beckwith (NT Professor at Wycliffe Hall, Oxford): “The last supper, as John describes it (and not just as the Synoptists do), has many of the peculiarities of the Passover meal. It is held in Jerusalem, despite the crowds, and not in Bethany (John 18:1; contrast 21:1f); it is apparently held later than the usual evening meal, going into the night (John 13:30); and the participants recline instead of sitting (John 13:2, 23, 25).” [Calendar and Chronology, Jewish and Christian: Biblical, Intertestamental and Patristic Studies (1996, Brill) 293.]
Jesus's disciples thought Judas was going to give to the poor at that time.1 This is relevant because “[i]t was customary to give alms to the poor on Passover night, the temple gates being left open from midnight on, allowing beggars to congregate there.” [Donald Carson, The Gospel According to John (W.B. Eerdmans, 1991), 475.]2
The meal was eaten “later than the usual evening meal, going into the night”.1 This is relevant because night “was an unusual time to be eating, unless it was so required.(Jeremias, Eucharistic Words, 44f.)” [“The Chronology of the Last Supper,” Westminster Theological Journal 53:1 (1991): 31.]
The Passover meal was not the next day.1
Roger Beckwith (NT Prof. at Wycliffe Hall, Oxford): The assumption of the disciples that what Jesus had sent Judas out from the Last Supper to "do quickly" might have been to buy necessities for the feast (John 13:27-30) is hardly intelligible if this was only the night following Nisan 13 and not that following Nisan 14, for it was custom in Judea to work on Nisan 14 until midday (M. Pesahim 4:1, 5). Nisan 15, on the other hand, was a solemn rest-day, in accordance with Lev. 23:5-8, so if this was the night following Nissan 14, Judas would already have been late in buying the necessities for the festival day and the Sabbath following, and needed to make haste. [Calendar and Chronology, Jewish and Christian: Biblical, Intertestamental and Patristic Studies (1996, Brill), 293.]
Donald Carson (NT Scholar & Prof.; Ph.d in Phil.): [If preparing for the Passover] why Jesus should send Judas out for purchases for a feast still twenty-four hours away. The next day would have left ample time. It is best to think of this taking place on the night of Passover, 15 Nisan. [The Gospel According to John (W.B. Eerdmans, 1991). 475.]
Colin Kruse (NT Scholar, Lecturer): [Some say Judas went to buy provisions for the following night, but] [t]his makes no sense, as there would have been opportunity to buy provisions the next day in that case, and there would have been no urgency. Possibly the disciples thought Judas was being sent to buy provisions, not for the Passover meal itself, but for the feast of unleavened bread which began at midnight. [John (Zondervan, 2003), 290.]
Barry Smith (Prof. of Phil. & Rel. Studies): If the meal had been held on the evening of Nisan 14, there would have been no need to buy goods that night during the meal, since there was still the entire next day to do such things. But if the meal was a Passover meal, eaten on the evening of Nisan 15, then the urgency would be understandable, since the next day was a high feast day, the Sabbath of Passover week. (Purchases were lawful during Passover night.)(Heawood, “The Time of the Last Supper,” 39f.; Jeremias, Eucharistic Words, 53.) [“The Chronology of the Last Supper,” Westminster Theological Journal 53:1 (1991): 32.]
The “high sabbath” was the day (6pm--6pm) just after the day of Jesus's meal-arrest-crucifixion.1
George Murrey (Prof. of NT at S.B. Th. Seminary): Bultmann pointed out that if the “high Sabbath” of 19:31 was the day of the sheaf offering, that would make it Nisan 16, which in turn implies that Jesus was crucified on Nisan 15, as represented in the synoptic Gospels (676 n.6). [Word Biblical Commentary: John (Word, Inc., 2002) 225.]
Roger Beckwith (NT Professor at Wycliffe Hall, Oxford): The statement of John 19:31 "the day of that Sabbath was a high day" is vague if it refers to Nisan 15 and not Nisan 16. John has already stated it seven times that they have come up to Jerusalem for the "Passover" (John 11:55, twice; 12:1; 13:1; 18:28, 39; 19:14). It would surely have been more natural, therefore, to say "the day of that Sabbath was the Passover" or "was the day when they ate the Passover", had he thought of it as such. If, however, it was Nisan 16, the day of the offering of the sheaf (Lev. 23:9-14), his expression is a very natural one. [Calendar and Chronology, Jewish and Christian: Biblical, Intertestamental and Patristic Studies (1996, Brill) 294.]
Sir William Smith, John Mee Fuller: Those, on the other hand, who identify the supper with the paschal meal, content that the special dignity of the day resulted from its being that on which the Omer was offered, and from which were reckoned the fifty days to Pentecost. One explanation of the terms seems to be as good as the other. [A Dictionary of the Bible, Vol 2 (William Clows and Sons, 1893) 721.]
“The Last Supper reported in John 13 precedes the Paschal Meal of the Jews. This is relevant because if it precedes the Passover meal, then trivially it is not the same as the Passover meal.”