This is one of the problems with mobile -- there's isn't any one universally correct answer to this (or most questions), so I'll answer to the best of my knowledge. IMSIs are what are associated with your identity (because it's your SIM and service) in a normal mobile plan, and it's what was (is?) used by carriers when they aggregate / analyze / sell location data.

IMEIs can be queried by a network core (not the tower) and US carriers probably do this every once in a while to check against their stolen phone database. It can be changed on some devices but not others. It's not inherently tied to you as a person but of course it is tied to that device.

For those who don't need mobile data service of any sort, I think that PGPP Relay does what's needed -- decouples your IP from your identity -- and you can use WiFi networks without revealing anything.