This is so awesome. The concept works so well and the variety shapes can be used for so many different things! In future levels you could even swap some of the shapes for even more possibilities.
I'm guessing it's probably to save dev time, but I really liked the "one big level" approach which let the player go their own non-linear path. I say keep it like that, you can still have "levels" like a normal puzzle game but have each level contain a few coins that you can pick in whatever order you'd like.
Only critique I have on the mechanics is that it did feel a bit janky when you were inbetween tiles, one puzzle could be solved by just spamming and hoping it lines up instead of doing the proper approach. But at the same time I like the fluidity of the movement, a grid based one maybe wouldn't feel as good. I guess you could make it so that the player smoothly "snaps" when you try to shapeshift, but otherwise have the movement as fluid as it is now.