Yes, I've observed the same with people who speak both Spanish and English.

As far as learning the culture goes, I think you can probably do that without paying a lot of money to learn the language. She can teach you a little bit here and there, enough to help you with the culture aspect. And language is only a part of culture. To learn it well, you have to live it. When you spend time with them, try to learn the culture by observation or ask her or her family and friends about things that seem culture-specific.

I just don't see the point of learning the language. First, they speak English, so there's no practical reason. Second, and I don't intend any disrespect, but you're just the boyfriend; you didn't indicate any kind of permanent relationship. Why spend the time and money if you may break-up in the near future? If I'm mistaken about that, then spending the time and money to learn the language is a very good idea.

It'd be a nice gesture on your part to learn Urdu, for sure. And if you really want to, do it. I just can't see any significant benefit from doing so.