It’s Spelled Yrian
Like engineers, café baristas aren’t particularly known for their outstanding spelling skills. They can spell most common names correctly, but when you tell them how to spell your name, you, the customer, are going to have a pretty bad time.
The OP clearly informed the Starbucks barista that their name was “Brian with a Y,” and that’s sort of what the OP got. Silly barista, you had no idea that his name is actually spelled Yrian. We all already know that Starbucks’ baristas are trained to misspell their customers’ names so we upload their mistakes on social media and promote their products for free. Outstanding troll, Starbucks.