It is known that where you learn sage mode affects it. Naruto and Jiriya both learned from the toads, and because of how it was learned on Mount Myoboku, they took on Toad characteristics. Naruto mastered it though, and as such only had his eyes transform, but Jiraiya took on many traits, from warts to frog like hands. Kabuto learned Sage mode from the Snakes in Ryuchi Cave, and as such took on Snake characteristics. Jugo's clan, or at least Jugo himself, Absorbs it naturally, and his personal side effect was the darker personality as he lost control.
As for Hashirama, we do not know where he learned Senjutsu, but the theory is that he mastered the Slug variant, since that is the third of the triple deadlock animals. However, Mitsuki seems to be able to use sage mode without having learned it from somewhere. This makes it most likely that if you learn sage mode from the Snakes, Toads, or some other animal that can teach you, you take on their traits, and mastery limits those traits to the eyes. If you learn it on your own, thanks to genetics or something else, you take on a unique transformation, like Mitsuki and Jugo. The best evidence for that theory is the curse mark, which is Jugo's method, but every user who transforms using it transforms differently. Again, that is a theory that just seems to fit somewhat.
Unfortunately, that is just about all we really got on it. Kishimoto did not go into full depth of quite a few things, and Sage mode is one of them. If we are lucky, then the Boruto manga might explain Mitsuki's Sage mode, and that should fill in a lot of the gaps.