These specific actions are definitely not part of the DMCA. In fact, it's basically the reverse. Unless you hire a lawyer to represent you, you must dox yourself to file a DMCA claim, which will involve handing over your name, address, and phone number to the platform committing the infringement against you, with the DMCA complaint requiring swearing under penalty of perjury that you are not falsifying any details.
> else you can [sue] the app store for not doing that.
This is, I think, the fantasy belief of someone who has never engaged with the legal system. You submit a notice of copyright infringement. They ignore it. Now what? Are you, as an independent developer, prepared to spend years of your life fighting to have it taken offline, out of pure spite, because you aren't going to get anything near the effort you put in? Even if you "win", you still lose, because it's just not worth it.
This is assuming you're even aware of the infringement. It was pure luck that I happened to discover the copyright infringement, in my case. It would be very easy for somebody to never discover that their game was re-labelled with a new name in a foreign app store. And once aware of it, actually trying to enforce my copyright quickly disabused me of the notion that copyright law could ever benefit individuals in any meaningful way.