Of the two:
- The dog's tail.
- The tail of the dog.
Grammatically, both are correct. However, the preferred is "The dog's tail."
A dog is an animal, therefore it holds possession of its tail thus making an apostrophe appropriate.
Also when writing, it is advantageous to communicate your point with as few words as possible making your point clear and concise. Hence, why "The dog's tail" is preferential in colloquial english as opposed to "The tail of the dog."
Lastly, if you would like to be super technical, the dog has only one tail and multiple legs. Each leg has a distinct orientation on the dog (front left, front right, hind left, hind right) as opposed to a table leg which may not show a difference between orientations.
This also depends on how specific you want to be. If you were telling someone that the "dog's leg is broken" and then they asked "which leg of the dog?" both of these phrases would be correct in their own right.