Looks like I missed this question last month, my apologies!
Spectre is specifically configured to make the copied password local-only currently, for security reasons.
Handoff is used set up in many different circumstances, and it may result in an unpredictable copying of your password to other computers, silently and without people realizing it. You can imagine for instance, a parent using Spectre on their phone and a child using the family computer, now able to paste the parents’ password.
I do see the added value, however. I’m thinking about how to best make this capability available in a safe manner.