Yeah, this is great advice. It's unfortunate when pressure to do social niceties holds you back from enjoying meaningful connection with people. Some people are better at remembering dates than others, and caring enough to ask (and put it in one's calendar) is at least as powerful a gesture as actually remembering it.

OP could take a jokey tone to things, such as by expressing surprise that she was able to remember, and then asking what her birthday is

You weren't supposed to because it's a joke.