In the way that you're thinking? No, it doesn't work like that. You can't take an elephant's trunk and put it on a giraffe, biology is a chemical pattern and mixing and matching requires far more effort than just "put that over there".
But in reality, we have transgenic lifeforms all the time, especially in plants, and we'll just have more in future. There's no market for meme mythological creatures, however. Not worth the sheer amount of effort it'd require to get right.