Why would anybody want these IDs? The simple answer is that app authors mostly make money by selling advertising, and advertisers like to know who's seeing their advertisements. The more app views they can tie to a single individual, the more they can track that user's response to different kinds of adverts and the more targeted (and, they hope, more profitable) the advertising towards that user. Using the same ID between multiple apps makes this easier, and so using a device-level ID rather than an app-level one is preferred. The IMEI is the most stable ID on Android devices, persisting even across factory resets.
The downside of using a device-level ID is, well, whoever has that data knows a lot about what you're running. That lets them tailor adverts to your tastes, but there are certainly circumstances where that could be embarrassing or even compromising. Using the IMEI for this is even worse, since it's also used for fundamental telephony functions - for instance, when a phone is reported stolen, its IMEI is added to a blacklist and networks will refuse to allow it to join. A sufficiently malicious person could potentially report your phone stolen and get it blocked by providing your IMEI. And phone networks are obviously able to track devices using them, so someone with enough access could figure out who you are from your app usage and then track you via your IMEI. But realistically, anyone with that level of access to the phone network could just identify you via other means. There's no reason to believe that this is part of a nefarious Chinese plot.
Is there anything you can do about this? On Android 6 and later, yes. Go to settings, hit apps, hit the gear menu in the top right, choose "App permissions" and scroll down to phone. Under there you'll see all apps that have permission to obtain this information, and you can turn them off. Doing so may cause some apps to crash or otherwise misbehave, whereas newer apps may simply ask for you to grant the permission again and refuse to do so if you don't.
Meitu isn't especially rare in this respect. Over 50% of the Android apps I have handy request your IMEI, although I haven't tracked what they all do with it. It's certainly something to be concerned about, but Meitu isn't especially rare here - there are big-name apps that do exactly the same thing. There's a legitimate question over whether Android should be making it so easy for apps to obtain this level of identifying information without more explicit informed consent from the user, but until Google do anything to make it more difficult, apps will continue making use of this information. Let's turn this into a conversation about user privacy online rather than blaming one specific example.