Nope.Yes, in the wacky world of the FAA and airplane can have any number of PICs, from zero to 900 (the maximum number of seats that can be stuffed into an A380).
Only one PIC at a time - the person who has final authority and responsibility for the operation and safety of the flight; has been designated as pilot in command before or during the flight; and holds appropriate category, class (and type if required) ratings. Period.
Logging is about writing things down after the flight is over.