In the Mirror of Erised scene, Harry walks around the school with the invisibility of his cloak and the warmth from the sweater Molly knitted for him. Christmas just happened, and both are the first genuine gifts of love that Harry had ever received since his parents died.
Harry asks Dumbledore what he saw in the Mirror, and Dumbledore said a nice pair of wool socks. Harry was perplexed that a man so interesting and powerful and wise would want something as dull as socks as his number one desire. After all, clothes are usually the least-liked gift for kids on Christmas, and socks are the dullest of clothes. This view is even held by Ron and Percy, who begrudge wearing the Weasley sweaters on Christmas. As someone who knits, I can tell you that knitting eight different sweaters for Christmas is pure magic.
I think Dumbledore saw this shirt, and remembered another family member who was stuck at home her entire life; a person who could have easily found knitting very soothing. After all, what else could an untrained witch like Ariana Dumbledore give her brothers for Christmas? The amount of love and thought put into each article of clothing must have been tremendous.
But Albus was a kid, one who felt like all kids when it comes to clothes. What's more, he was embarrassed by her very existence. He probably never wanted anything to do with anything she made. He may have even tossed it after getting it.
But it's Christmas once again, and Aberforth, despite living in Hogsmeade, wasn't at the feast. I think he's always seen Ariana when he looks in the mirror, but it wasn't until Christmas that the one thing he saw was his family together and him getting just one pair of the lovely socks she made as a kid.