The oldest game I can think of that used this perspective is the first Splinter Cell on the PC. When you wanted to shoot your pistol, the camera moved over Sam's shoulder and then a reticle appeared.

Do you remember any older games using this? Specifically shooters?

According to the list of games associated with this concept, it would be Conker's Bad Fur Day.  I never played it so I can't really verify that.

I'm not sure, but it was probably earlier than Splinter Cell, there must've been something PS1 era.

I know that Conker's remake on the original Xbox had over-the-shoulder shooting, but just like you, I've never played the original to know.

I seem to remember Tomorrow Never Dies on PS1 had an Over-The-Shoulder perceptive.
 
I can't be sure though.

Syphon Filter was an over the shoulder camera if I remember correctly.

I think it would also have to be splinter cell when it came to shooting your weapon. But the first one that I've noticed to use it the entire game would have to be Resident Evil 4.

No idea which one was first, but I´m pretty sure Resident Evil 4 popularised it.