Edit: I can now paste on eBay (not sure what went wrong the first time) but PayPal is still actively preventing pasting a new password.
Perhaps my region (UK) still uses the old password change page? For clarification, I'm using the change password function once logged in and not doing a forgotten password reset.
This doesn't let me paste, giving the aforementioned tip that I should copy/paste.
However on PayPal, when pasting I received a little tooltip-style popup saying something along the lines of "Please do not copy and paste passwords.", followed by their password criteria.
Pasting into other fields (including the login page password field) worked perfectly fine.