Not aware of any official back story on the lizard, but Paul Starkey (Trek BBS reg Starkers) wrote "Good Luck Charm" which explains it pretty darn well
Beautiful fan story. Now, we just want to learn why Starfleet's ship surgeons have this strange hobby of collecting (apparently animal) skulls (compare redressed doctor's office aboard the Defiant
Come to think about that, that's probably what they are, "Terran" animal skulls because that would have been cheapest for the actual production. Thus, if we were able to figure out to which animal a certain skull belongs, we would know what these skulls in McCoy's wall shelf actually are and what they should look like...
And this could also explain why they put the "scary skull" poster over the Season Three Office desk, maybe to convey the illusion the ones in the wall shelf are alien too.
P.S. Those additional screencaps with the table are great, Donny. Unfortunately they didn't always attach the thinner table leg to the wall.