Trying to find information on Android emulators on Linux is a pain in the ass and I spent like three hours trying to find out what I needed to know to make it work so I'm going to put this on some page that isn't lost in some discord server even though it's still going to get drowned out by SEO garbage. As a side-note, the end result of this is just running an Android install in a window, so any problems with keyboard/mouse input that show up on a regular Android phone are going to show up here. I mention this because Fate/Grand Order only accepts touch inputs and mouse inputs don't work on a regular phone, and that problem was replicated in the emulator, because again it's just running Android in a box.
I ended up using Waydroid. Anbox was apparently the old preferred Android emulator but development on it has stopped, so use Waydroid. Follow the instructions in the link to get it installed. It only works if you're using the Wayland desktop compositor, so you should switch to that if it doesn't launch anything after you install it. Additionally, when the install window pops up you're going to want to pick the GAPPS version and not VANILLA. Vanilla is a basic Android install without the Play Store or any of the Google apps, which makes it a real pain in the rear if you want to use anything that relies on Google Play services, which is most of the things I would personally want to use an Android emulator for.
After installing it, to use the Play Store you need to make it Google Play certified. This page in the docs explains how to get the device ID that you put into the Android certification page. Do that and you can download apps off the Play Store onto your computer. Great. Now unfortunately those apps are probably going to be ARM-based and not x86-64, or at least Another Eden was and didn't work until I went and did the next bit.
To get ARM-based apps running on the emulator I followed the advice I found in this thread and installed libndk from this Github Python script. So far the emulator seems to work, or at least not immediately crash apps on boot like it was doing before, so I figured I might as well put this write-up up. The page will be expanded if I run into more problems.
Also, as with most Android APK-related problems, if Google Play doesn't work then APKPure generally works for getting some specific regional version of an app. Do be careful and check the version number first though. APKs are weird and won't let you revert back to an older version number if something goes wrong, which it did one time for me where some regional japanese phone carrier's app store screwed up the version number for Fate/Grand Order to some version like ten thousand units ahead and that got propagated into APKPure's database and nearly killed like five of my friends' accounts. So, y'know, check first.
Alright great, you've booted up a game and for some reason it's acting weird with mouse input. I've seen this in FGO and Brown Dust 2, and none of the google searches I did gave me anything useful. This is because sometimes games don't have any mouse handling, since I mean who uses a mouse on a phone? To fix this we'll need to simulate touch inputs. Looking at the docs on this page seems like it should do what we want, and it does. For the case of FGO, entering waydroid prop set persist.waydroid.fake_touch com.aniplex.* into a terminal and then restarting Waydroid will get it working. If you're trying to fake touch input in another game that's having issues with mouse input, you'll want to use an app like Zarchiver to go into the filesystem and navigate to /storage/emulated/0/data and look for the game's folder to get the name to enter in. It's usually going to something like like com.[insert publisher name].[game name], but not always.
My Steam Deck uses x11 for its desktop compositing and is pretty much stock so I will go with a tentative "no*******, for now", and we will be ignoring the many asterisks attached to anything to do with Linux. Sure, it's probably technically possible to set up Wayland to work on the Deck but right now it's just a bit too much of a pain in the ass to bother. I tried a couple other scripts I found to install an Android emulator on the Deck and none of them worked. I'll update this section if an update to the Steam Deck changes the desktop compositor.