Use CSS. There are lots of good CSS tutorials around the web.
Could you kindly provide some links?
I’ve explored numerous tutorials, but perhaps my choice of keywords isn’t quite right for what I’m seeking.
Actually, my first answer was wrong.
The main problem here is that the taxonomy list dropdown is an option tag, and pretty much the only things you can style are the colours. Usually when people do custom-styled dropdowns, the select and option tags need to be swapped with something else that can be styled. That’s a bit more complex, though, and is not really a Relevanssi support question but a general web dev question.
Ah OK.
Is the loading animation also impossible to change?
Yes, I suppose so, unless you modify it a lot. But this is really outside my expertise. You’ll get better answer for this from people who are CSS experts. The question isn’t restricted to Relevanssi or even WordPress, so asking around elsewhere will get you better results.