What is the name of Google Translate's (android) functionality, that with the app I can search for any highlighted text? I can highlight any text in any app, a little list comes up, and shows me that the text can be "used within the Google Translates app" - in this case the text will be translated. See screenshot below. "Fordítás" means "translate" in my language.  What Android system feature is allowing this behavior?
In this particular case you are using Google Keyboard which has the "Translate" feature. Such a feature is built into the custom keyboard.

Using Google Keyboard you can see the "Translate" menu option:

Google Keyboard Cut Paste Menu in Chrome:



Other keyboards which can come as default on devices like Samsung have no such feature.

Samsung Keyboard Cut Paste Menu in Chrome:



It is also possible that a custom item will appear in the pop-up menu which would be specific to that particular app. So, if you want to have that feature (or possibly others), you'll need to check if its feature of the app or the custom keyboard.