"On my Mac and Apple Watch, when I connect and ask it to show it on Find My it doesn't come up with a device. Is it possible if I sign out of my friends iCloud account on their phone and sign into mine to then add them to Find My?"
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No. I recommend you let your friend's phone be, by keeping off of it. But, that's THEIR call. If you were to log into it, they'd have to wait 90 data to get back into it on their own account. If this is on a device that is assigned to THEIR Apple Account --then that explains all of this. They should be seeing this under their Apple Account, despite them being yours.