Not super common in software, I know people in law and a few other fields that always have to do it...
The background check thing feels mostly like a boogie man to me. It's convenient for recruiters that engineers believe that, but I've never seen it happen in practice. And if it did (how does this even manifest? they require you to send them a paystub? they can't compel your previous employer to divulge this info), you should be outraged! It's none of their business what your paycheck at your previous employer looked like or contained. I would respectfully decline to provide this information, and if they insisted I would edit or photoshop it, and give serious thought about whether this is the type of employer you want to work for. You should be indignant about the fact that an entity you are planning to devote the majority of your waking life to for the next few years is comfortable with systematically fucking you over like this.
I don't normally like to lie, but I see no moral issue with it in a situation like this where the system is intentionally rigged against you. They are systematically depriving people of pay; I'm not going to lose sleep over a small lie that rights that wrong.