I actually ran in to the same issue last night. My wife's iPhone was using my apple id as well (for ease of sharing all pictures), but with now adding her watch to the mix, it's a cluster, so I just set her up with her own apple id, but added her into the family sharing account.
I wish Apple gave you an option to automatically share all pictures (like when using the same apple id), rather than to manually uploading each photo you wish to share.
Also, I have an apple watch, so when i initially setup her watch i just used my apple id, but then after it had setup, i changed the name of the watch, but under the activity app, my named showed up twice.
Yes, your watch is tied to your phone, so under the watch settings app on your phone, under general, then scroll down to Apple ID, is where you would change it
