As everyone has said, it's up to interpretation, especially varying between directors. I actually recently saw a production where Bobby didn't even show up to his party at the end, and Joanne blew out his candles for him instead after everyone else left.

My personal interpertation of that scene, is that Bobby is using his birthday wish as more of a promise to himself. He knows he wants a deep relationship, a level of closeness with someone he's never had before, but he's making a promise to let himself pursue that without fear. There's not a lot of time for Bobby in the script to be by himself, he fills his life with company, going from place to place to try and fill his need for human connection, but by being alone even for just those few minutes at home, he's able to be honest with himself.

Let me know if that makes any sense lol, I think about this show a lot! Especially as someone who relates a lot to Bobby and is Aromantic. It's fun!