Yes and no, the skin connects to the spinal nerve in regions known as dermatomes, there being 31 spinal nerves and 30 dermatomes, with left and right sides. Some nasty rashes get confined on a dermatome basis, to not cross the left-right axis, this being because the nerve roots are the problem, rather than the skin, per se.

Hence, you have a pain map of sorts, based on things such as the dermatomes. Utility of the skin also comes to mind, the hands, although 'tough' have a lot of nerves and, allegedly, a job stopper hand tattoo will be extremely painful. Same with the chest, which is 'underbelly' and not 'defended', whereas your shoulder will not have the same 'evolutionary' choices.

Coupled with that is the evolving technology, in the olden days, tattoo machines looked like they were held together with rubber bands, and people swore by these contraptions. Nowadays you have something more like a glorified electric toothbrush, with most artists just preferring the modern tools, that also hurt less. Aftercare has improved too.

Different artists will also impose a different level of pain. Pain also is 'in the eye of the beholder', for your first rodeo you probably had a certain amount of adrenaline going round, for the 'other shoulder' you won't have this 'natural pain killer' to quite the same abundance, so it might hurt more than you expect, rather than less.

Nonetheless, dermatomes, you need to go down that rabbit hole! I never knew of the things until recently.