New iPhone - transfer AirTag items from old phone to new phone

Hi, I've done the following

1. Setup my new iPhone 13 with the standard setup procedure, including copying data from old to new iPhone by keeping the devices close to each other (bluetooth data transfer?). Everything worked smooth as usual,, apps etc
2. Noticed that my AirTag items were not copied across
3. Noticed that on my new iPone I keep getting the notifications "AirTag Found Moving With You"

My understanding is that this notification is shown when an airtag tag you don't have in My Items is in close proximity to your phone (bluetooth range).

Questions:

• Is there a procedure for how to transfer AirTags to a newly bought phone. Or do you have to setup everything from scratch again? Note! my AirTags were all setup with the same Apple ID as I use on my new phone, and I still have my old phone to confirm etc
• I read somewhere that you should not get these privacy notification if the AirTag is associated with the same Apple ID as you have on your phone. But I do get them, despite the fact that I', signed in with the same Apple ID. Is there someplace onm my new iPhone where I forgot to sign in?

Note! My old iPhone shows up in my device list as "iPhone (2)"

I'm really surprised that there seem to be so little information about this procedure, for such a common scenario. I.e. getting a new phone

Thanks