"On the left: Benedict Cumberbatch in costume. On the right: Benedict Cumberbatch in character. Such a subtle but amazing difference. Benedict and Sherlock are completely separate people as far as I'm concerned. It's truly amazing.
Sherlock gave Mycroft this tie as a joke one year. Mycroft was so touched that Sherlock gave him anything, so he wears it with pride even though Anthea mocks him for it <--- headcanon accepted. Or maybe he just likes umbrellas that much.