I keep being told this is a conspiracy and that several people checked on this and found no evidence of them spying on you, but I still have a hard time believing they're not doing something really sketchy.

The case that was the final straw for me was when I was chatting with my partner and remembered a funny song from my childhood, so I opened YouTube on Safari and showed it to her. A couple of minutes later, she opens Instagram (on her own phone) and the first "follow suggestion" is the artist from the song. She had never heard of this song before, much less of the artist (which is not famous at all).

I would understand if everything happened on my phone/accounts, but the suggestion was on her phone and account. I don't think they're literally listening to you, but there's definitely a GPS-based user relationship table somewhere which reflects what you do to everyone they think has some connection to you and is physically close to you.