The phrase in question is «Ἰησοῦν τὸν λεγόμενον Χριστόν». It does not mean that Pilate himself was confessing Jesus to be “the Christ.” Rather, the expression is commonly used in the NT to describe what people are generally called or named by others.1 For example,
Ἰησοῦς ὁ λεγόμενος Χριστός — Jesus who is called Christ
Σίμωνα τὸν λεγόμενον Πέτρον — Simon who is called Peter
Σίμων ὁ λεγόμενος Πέτρος — Simon who is called Peter
τοῦ ἀρχιερέως τοῦ λεγομένου Καϊάφα — of the high priest who is called Caipahas
εἷς τῶν δώδεκα ὁ λεγόμενος Ἰούδας — one of the twelve, who is called Judas
εἰς χωρίον λεγόμενον Γεθσημανῆ — unto a place called Gethsemane
δέσμιον ἐπίσημον λεγόμενον Βαραββᾶν — a notable prisoner called Barabbas
εἰς τόπον λεγόμενον Γολγοθᾶ — unto a place called Golgotha
There are countless other examples in the NT. Thayer on λέγω states the following:2
e. to call by a name, to call, name; equivalent to καλῷ τινα with the accusative of predicate: τί με λέγεις ἀγαθόν; Mark 10:18; Luke 18:19; add, Mark 12:37; John 5:18; John 15:15; Acts 10:28; (1 Corinthians 12:3 R G); Revelation 2:20; passive with predicate nominative: Matthew 13:55; 1 Corinthians 8:5; Ephesians 2:11; 2 Thessalonians 2:4; Hebrews 11:24; ὁ λεγόμενος, with predicate nominative he that is surnamed, Matthew 1:16 (so Matthew 27:17); Matthew 10:2; John 20:24; Colossians 4:11; he that is named: Matthew 9:9; Matthew 26:3, 14; Matthew 27:16; Mark 15:7; Luke 22:47; John 9:11; cf. Fritzsche on Matthew, p. 31f; of things, places, cities, etc.: τό ὄνομα λέγεται, Revelation 8:11; participle called, Matthew 2:23; Matthew 26:36; Matthew 27:33; John 4:5; John 11:54; John 19:13; Acts 3:2; Acts 6:9; Hebrews 9:3; with Ἑβραϊστί added, John 19:13, 17; (cf. John 5:2 Tdf.); applied to foreign words translated into Greek, in the sense that is: Matthew 27:33; John 4:25; John 11:16; John 21:2; also ὁ λέγεται, John 20:16; ὁ λέγεται ἑρμηνευόμενον (L Tr WH μεθερμηνευόμενον), John 1:38(39); διερμηνευομένη λέγεται, Acts 9:36.
1 It does not exclude the speaker from also calling the person by the name, but it certainly does not require it.
2 Thayer, p. 375, λέγω, II., 2., e.
Wilke, Christian Gottlob. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Trans. Thayer, Joseph Henry. Ed. Grimm, Carl Ludwig Wilibald. Rev. ed. New York: American Book, 1889.