– How common is it for different developers to end up building similar games?
It happens all the time. Sometimes you don't realize it and think you have a new and unique idea, but often that idea is inspired by something external—an event, a movie, or an old game you knew. Nowadays, there are many people who were exposed to the same inspiration and are also creating games.
Many famous games aren't usually the first ones. For example, if you think of platformers where you jump on your enemies, the first thing you probably think of is Super Mario Bros. for the NES. But if you investigate, you'll see that many concepts and mechanics predate it. There were other platformers and others where you killed your enemies by jumping on them.
This doesn't mean that Super Mario copied those games; rather, that the "seed" isn't really as unique as you initially thought.
– Do people usually stop when that happens, or keep going anyway?
A game is unique not only because of its core mechanics or setting, but also because of all its components: art, music, production, etc.
Today, there are many games that are similar and yet completely different.
Without more details, I couldn't tell you how unique or common your idea is, or if what you thought is actually the same as what you found, as that depends largely on the details, not just the general idea.
– What’s considered respectful in this kind of situation?
The first thing you should do is play the other game and make sure it's actually the same as what you had in mind.
As I mentioned above, sometimes a game can seem similar, but be different at the same time.
For example, maybe they focus on X mechanic, and you really want the focus of your game to be on Y. Even though they share the same mechanics, there's plenty of room for them to be different.
For example, Magic the Gatering vs. Yu-Gi-Ho.
I remind you that I'm speaking in general terms since you didn't go into details.