Of course God can do things any way He wants. What is interesting is that when He tells us how He did it, so many don't believe Him!
And yes, variation is easily established. However you will note that no matter how many variations of dogs, or fish, or bacteria, or whatever we come up with, they remain dogs, fish, bacteria, and so on.
Variations off the original created populations are totally supported by what we know in genetics and biology. Massive form changes, however, are totally unsupported and are actually contradicted by what we know in genetics and biology.