Parmandur
Book-Friend, he/him
No, that's not accurate to 5E D&D, the rulebooks contradict that directly.Flavor bakes in.
Indeed, if the players want to reflavor a spell, they need to completely rewrite the spell. It is a new spell. Then I as a DM use the flavor of that new spell description to determine any implications within a given encounter. If a spell creates chickens, perhaps that makes the dogs go crazy. Depends on the dog.