Would it be possible to analyze the popularity of specific tags on e621 over time? Or even the popularity of specific tags of all time? Using upvotes/favorites given to posts with a given tag in a specific time period. Is this information saved anywhere?

https://e621.net/db_export has the upvotes on any given post, that's as close as you're going to get

Yeah, we don't have time-domain information on that, at least not without using past exports to calculate it. (Only recent exports are available, though)

http://web.archive.org/web/*/https://e621.net/db_export/* Only goes back 2 years but it does what you want. You can also try looking for archived search results (post result pages) but that is not likely to work.