It’s the same metaphor—you’re signing the (virtual) logbook to get in.
Another reason I prefer sign in is because there's less discord about its usage. I have never seen anyone try to write signin. Maybe they'll hyphenate it, but even then people seem less inclined to use sign-in as a verb compared to login.
- He will log in to the system.
- What's your login?
- Please sign in.
- That's the sign-in table over there.
Along these lines, does the button/link describe the action (click the button to log in) or the type of button (the login button).
The question of whether to to use sign in or log in is separate.
I go back and forth on prescriptivist vs descriptivist versions of language, though at the end of the day, languages that are in use change. I think it makes sense to create the equivalent of a stylebook for a publication or site so that there's consistent usage (prescriptivist), but understand that there are going to be inconsistencies across the use of language as a whole (descriptivist).