Obligatory Caveat: These are headcanons pertaining mostly to Wesleyverse.
Sauron’s relationship with the werewolves is a working relationship. I think when he’s first sent to besiege Tol Sirion with a bunch of wolfy reserve troops, he’s actually SUPER resentful.
I mean, he’s been working in Angband, at the right hand of a Vala, with his own forge and command of central operations, and suddenly it’s like “GOOD NEWS, YOU’RE BEING PROMOTED TO LORD OF THIS BACKWATER ELF CASTLE IN THE MIDDLE OF A SWAMP! 😀 Here are your new subjects, the wolf-people. Please wear your flea collar at all times.”
That can’t have felt like a step up in the world.
I imagine that there were werewolves native to that area anyway, wild ones that had been living there for a long time, never making themselves a target for extermination by Elves. They are a useful resource for Melkor to utilize for the takeover of Tol Sirion because if he can leverage their help, he can afford to move fewer troops out of the North. They’re also much faster, scarier, and more maneuverable than Orcs. BUT they require special handling– hence the necessity to put a reliable Maia on the job.
Needless to say, the job, and the werewolves, grow on Sauron/Thû over time, to the point where he’s comfortable taking a wolf shape and running around as part of the pack.
I also think he gets used to having To-en-Gaurhoth as his personal bachelor pad; just him and the puppies and Thuringwethil and some Orcs, throwing house parties and letting it all hang out. It’s a friendly working relationship. Thû and his wolves are pretty tight, and he identifies with woflish things from then on.
Melkor’s relationship with his dragons is a little different in that they are his actual children. He is both father and mother of dragons, made with his unique and slightly unsettling demiurgical reproductive abilities. No, he doesn’t have Eru’s Secret Fire, but he does have his –own– fire, a LOT of it, and especially in the early days when he felt infinite and invincible, he parceled that out like cheap candy at Halloween to his various monster babies.
He adores them and spoils them and treats them like special snowflakes that can do no wrong. I mean, they’re basically shiny, scaly clones of himself, and Melkor LOVES himself. And dragons love THEMSELVES. And they love Melkor. And they all loooove each other– and that’s cuz none of them got enough love in their childhoods. And that’s showbiz. Kid.