Nor do the airlines, but they seem to think they can define what is PIC and what isn't. Besides, the FAA bends to the insurance industry all the time; hence the note in the FARs allowing multi solo PIC time to be logged with an instructor aboard.
the airlines dont need to. Part 121 requires among other things, multi engine aircraft, at least a 2 pilot crew (more in certain cases) and flight attendants to operate.
The PIC/SIC issue i think you are getting at is also not defined by the airlines. "PIC" has 2 definitions. In 14 CFR 1, PIC is defined as
14 CFR 1 said:
Pilot in command means the person who:
(1) Has the final authority and responsibility for the operation and safety of the flight;
(2) Has been designated as the pilot in command before or during the flight; and
(3) Holds the appropriate category, class, and type rating, if appropriate, for the conduct of the flight.
14 CFR 61 also deals with defining PIC and further makes the distinction between "acting" as a PIC and "logging" PIC. I'm not going to delve into that distinction, you can read about it elsewhere, in depth.
To act as a PIC in a 121 environment requires everything that part 61 requires of you in your Cessna 172, and in addition various 121 regs are brought into the picture. Some of the highlights are 121.440, required a PIC to have had a line check within the preceding 12 months; 121.441 requiring simulator proficiency checks or simulator training within the last 12 and 6 months; 121.443 and 445 which detail requirements of PIC's operating into specific environments. All this is in addition to the basic crew member requirements spelled out in part 121.
Now im not entirely sure what exactly the airlines "think they can define what is PIC and what isn't", but i'd be happy to take a stab at it if you can clarify for me. Is this about FO's logging PIC (part 61) time? Airlines not accepting shady kingair or emb110 part 91 SIC time? something else?
To the second part, yes you have a point about multi engine solo time, but i wouldnt be surprised if someone like AOPA had a hand in that, making life a little easier for everyone doing multi-engine training. That situation is pretty limited in scope though, only dealing with "solo" multi-engine requirements pre-certification, and i believe then only with the satisfying of solo/PIC requirements for specific XC flights. That doesnt mean you can hop into the right seat of a part 91 multi-engine airplane and log SIC because your "insurance" or "boss" requires 2 pilots.
If you are going to log SIC, do it right. Some situations allowing you to write in that column include specific safety pilot agreements (PIC/PIC vs PIC/SIC, you can do it both ways), part 91 on an aircraft whose type certificate requires a second pilot, part 135 with the proper training and checkrides, part 121 with the proper training and checkrides.