Who do Catholics say the Mary in Matthew 27:56 is?

The standard answer is that the Greek word "adelphos" can be translated as "brother", but can also mean "cousin" or having the same national origen.

a brother, whether born of the same two parents or only of the same father or mother having the same national ancestor, belonging to the same people, or countryman any fellow or man a fellow believer, united to another by the bond of affection an associate in employment or office brethren in Christ his brothers by blood all men apostles Christians, as those who are exalted to the same heavenly place -outline-

For Catholics, they understand the word to mean cousins (different parents), while Orthodox understand it to mean "step-brothers" with Joseph having children prior to marrying Mary. Thayer's using Strongs believe it neither, but only brother (same mother, different fathers).

STRONGS NT 80: ἀδελφός ἀδελφός, -οῦ, ὁ (from α copulative and δελφύς, from the same womb; cf. ἀγάστωρ) [from Homer down];

a brother (whether born of the same two parents, or only of the same father or the same mother): Matthew 1:2; Matthew 4:18, and often. That 'the brethren of Jesus,' Matthew 12:46, 47 [but WH only in marginal reading]; Matthew 13:55f; Mark 6:3 (in the last two passages also sisters); Luke 8:19; John 2:12; John 7:3; Acts 1:14; Galatians 1:19; 1 Corinthians 9:5, are neither sons of Joseph by a wife married before Mary (which is the account in the Apocryphal Gospels [cf. Thilo, Cod. Apocr. N. T. i. 362f]), nor cousins, the children of Alphæus or Cleophas [i. e. Clopas] -ibid- Vines understands it this way.

Brother, Brethren, Brotherhood, Brotherly: denotes "a brother, or near kinsman;" in the plural, "a community based on identity of origin or life." It is used of: (1) male children of the same parents, Matt 1:2; 14:3; (2) male descendants of the same parents, Act 7:23, 26; Hbr 7:5; (3) male children of the same mother, Mat 13:55; 1Cr 9:5; Gal 1:19; (4) people of the same nationality, Act 3:17, 22; Rom 9:3. With "men" (aner, "male"), prefixed, it is used in addresses only, Act 2:29, 37, etc.; (5) any man, a neighbor, Luk 10:29; Mat 5:22; 7:3; (6) persons united by a common interest, Mat 5:47; (7) persons united by a common calling, Rev 22:9; (8) mankind, Mat 25:40; Hbr 2:17; (9) the disciples, and so, by implication, all believers, Mat 28:10; Jhn 20:17; (10) believers, apart from sex, Mat 23:8; Act 1:15; Rom 1:13; 1Th 1:4; Rev 19:10 (the word "sisters" is used of believers, only in 1Ti 5:2); (11) believers, with aner, "male," prefixed, and with "or sister" added, 1Cr 7:14 (RV), 15; Jam 2:15, male as distinct from female, Act 1:16; 15:7, 13, but not Act 6:3. * [* From Notes on Thessalonians, by Hogg and Vine, p. 32.] Notes: (1) Associated words are adelphotes, primarily, "a brotherly relation," and so, the community possessed of this relation, "a brotherhood," 1Pe 2:17 (see 5:9, marg.); philadelphos, (phileo, "to love," and adelphos), "fond of one's brethren," 1Pe 3:8; "loving as brethren," RV; philadelphia, "brotherly love," Rom 12:10; 1Th 4:9; Hbr 13:1; "love of the brethren," 1Pe 1:22; 2Pe 1:7, RV; pseudadelphos, "false brethren," 2Cr 11:26; Gal 2:4. (2) In Luk 6:16; Act 1:13, the RV has "son," for AV, "brother." -ibid-

So, many believe the word adelphos may mean not only uterine siblings, but also of the same nationality, like cousins or step-brothers and step-sisters.

To answer the last question, it may appear there is a contradiction between Catholicism's Tradition and Scripture, but that is not the same as saying the Bible contradicts itself.

When answering Bible questions we always have to let the Bible answer the Bible. Nowhere in scripture did it state that Mary remained a virgin for life. In fact scripture states in Matthew 1:25, speaking of Joseph, and knew her not till she had brought forth her first born son. The word knew in scriptural terms means to have sexual relations with. The NLT says, But he (Joseph) did not have sexual relations with her until her son was born. It was very important that Mary was a virgin before the conception of Jesus seeing that his birth was from God and not by man. After Jesus was born was irrelevant because the act had already manifested. Also for Mary and Joseph to live a life as virgins is irrelevant also.

The accepted answer succinctly nails it. To add a bit more information regarding it from the bible:

In the Gospel of John, we see that there are three people named Mary standing by the cross: the mother of Jesus, Mary Magdalene and Mary of Clopas. Furthermore, John tells us that Mary the mother of Jesus had a sister named Mary of Clopas (who was thus Jesus' aunt).

John 19:25: "but standing by the cross of Jesus were his mother and his mother's sister [adelphe], Mary of Clopas, and Mary Magdalene."

In the Gospel of Mark, we see a group of women looking on the cross from afar off, two of which are identified as Mary Magdalen and Mary the mother of James the less and of Joseph.

Mark 15:40: "And there were also women looking on afar off: among whom was Mary Magdalen, and Mary the mother of James the less and of Joseph, and Salome" [also referenced in Mark 15:47 and Mark 16:1]

In the Gospel of Matthew, we see a group of women looking on the cross from a distance, and three of which are identified as Mary Magdalene, Mary the mother of James and Joseph, and the mother of the sons of Zebedee.

Matthew 27:55-56: "There were also many women there, looking on from a distance, who had followed Jesus from Galilee, ministering to him, among whom were Mary Magdalene and Mary the mother of James and Joseph and the mother of the sons of Zebedee." [referred to as “the other Mary” in Matthew 27:61, Matthew 28:1]

Unifying these three gospel accounts, we see Mark and Matthew identify a Mary as the mother of James the less and Joseph. John records three people named Mary by the cross, and the only Mary of those three who would be fit to identify that way would be Mary of Clopas.

Thus, Mary of Clopas is identified as Mary the mother of James the less and Joseph. James the less and Joseph are thus cousins of Jesus. Jerome (383 AD) expands this argument to conclude that the adelphos of Christ named as James, Joseph, Simon and Judas (Matthew 13:55) are all four his cousins, likely children of Mary of Clopas.

If interested, this wiki breaks out that family tree in more detail, supplementing it with historical records.