Indeed, cousins were sometimes referred to as brothers and sisters, although usually not in conjunction with a reference to a parent. In other words, it was common to introduce your cousin as your brother, but not to refer simultaneously to your mother and cousin as mother and brother. So when we read in Matthew:
55 "Is not this the carpenter's son? Is not His mother called Mary, and His brothers, James and Joseph and Simon and Judas? 56 "And His sisters, are they not all with us? Where then did this man get all these things?" (Matt. 13:55-56, NASB)we find no reason to believe that brothers and sisters means cousins. This is similar to this verse:
"If anyone comes to Me, and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be My disciple. (Luke 14:26)in which, once again due to the simultaneous references to parents, it is quite dubious to hold that brothers and sisters are actually cousins.
In addition to parallel accounts, scripture refers to Jesus' brothers in other passages:
Then His mother and His brothers arrived, and standing outside they sent word to Him and called Him. (Mark 3:31, NASB)These references, taken together with the fact that the Bible never teaches of Mary’s perpetual virginity, paint a clear picture that those spoken of are Jesus' actual half brothers and sisters. The only contrary scriptural argument is weak, namely that it is never stated explicitly that "brother means brother, not cousin or close relative."
But I did not see any other of the apostles except James, the Lord's brother. (Gal. 1:19, NASB)
After this He went down to Capernaum, He and His mother and His brothers and His disciples; and they stayed there a few days. (John 2:12, NASB)
These all with one mind were continually devoting themselves to prayer, along with the women, and Mary the mother of Jesus, and with His brothers. (Acts 1:14, NASB)
The fact that Jesus asked John to care for his mother is a reasonable counterpoint. If Mary had other sons, wouldn’t they be responsible for her care? Legally, yes. But it must be remembered that at the time He spoke these words (John 19:26-27) it is very likely that none of Jesus' brothers were believers.
For not even His brothers were believing in Him. (John 7:5, NASB)(Given that they were dishonoring Jesus, it seems even more unlikely that John would refer to Jesus' cousins as brothers. Also, Jesus' half-brother James was converted either as an eyewitness to the crucifixion or after the resurrection (1Cor 15:7).)
We have no details regarding the condition of His brothers' households at the time of His death. However it is easy for us to imagine a situation wherein we would prefer that a beloved fellow believer care for our mother rather than a sibling.
To all of this one must add the clear implication of:
But he [Joseph] had no union with her [Mary] until she gave birth to a son. (Matt. 1:25, NASB)