I had to do a fresh install of my website due to some recent issues I've been having. I still have the old site running and will delete it once I get everything transferred over.

Where, specifically, are the menu items stored? I need to copy the contents and do a manual import to the new site. Re-creating the entire menu manually is possible but will just take forever, since my menu consists of 130+ items all with custom details and icons.

Correct me if I'm wrong, but I'm pretty sure they're stored in a file instead of the database because when I did a fresh install, I overwrote all the new files with the old ones in an effort to restore my website (I was having MySQL issues, so I tried using the old files but with a freshly installed database). This seemed to overwrite the default menu with the one I created.

No, just menu_items table.