Usually they’re talking directly to you in a way other photos and videos aren’t.
They’re the kind of thing you want to save forever.
I can save a few voicemails by emailing them to myself or backing them up individually - but is there a method for bulk backing up voicemails to a cloud backup provider away from your carrier? My Google fu seems to say no so far..
Another option would be to set up your own automated phone system, and program that to navigate through and process your voicemail. You might have to go through the analogue hole (that is, record the playback rather than have the message file itself sent to you), but you might be able to process a large number of messages in that way.
StackExchange or a similar site might also have more suggestions.
Interesting question, FWIW.
(If it's not obvious, I've no direct experience of relevance here. I did have a workplace voicemail system that would send messages as files at one point, but that was long, long ago.)