Android - How can I customize the list of services that pops up when I choose to "share" something in an app?

The items in the list are determined by which installed apps support the ACTION_SEND intent. Any app that supports that intent will be included in the list. I don't think there is a way for users to modify the list other than installing/uninstalling applications.

I personally use AppChooser Pro for about a year already (there's also a free version to test first if you like it). It's quite similar to Andmade Share which already was recommended with an older answer here, but first looks better (OK, that's quite subjective), but for me also works better, somehow more "intelligent":

^{AppChooser: List and grid view (source: Google Play; click images for larger variants)}

First, it offers a list and a grid view – as above screenshots show. You can switch between both using the appropriate icon on the bottom (in the middle; easy to identify in the screenshots). You can sort the list alphabetically, hide items you never (or rarely) use while still being able to show them “ALL” (appropriate icon again), select multiple targets, and more. By default, the list/grid order is kind of LRU: once you used a target, it will be moved to the top.

I immediately went for the paid version (not even taking a look at the free one), and never regret it. Differences between the two are listed on the Playstore page of the free version.

Unfortunately, this app has been removed from the play store by the developer.

You can't change the list, short of removing the apps in question. Speaking as an app developer, I can say it's also hard to do it from that side too - so we're pretty much stuck with how the system works.

It may be annoying, but the system makes interop so easy between applications, both from the user and developer perspectives, that it's hard to complain.