Do you have different pet names for each partner?
I'm renown for giving nicknames... some more desirable than others ~grins~
I do have different pet names for each partner. These are based very much on individual contexts, so I don't use them interchangeably. However, I will also call most partners and friends "gorgeous", "darling", etc. My GF and I find the term "baby" quite intimate. She only uses that with myself and her hubby, and I only use it with her and my best friend.
Would you feel slighted if your love used the same pet name for all partners?
Hmmm... That would probably depend on various factors. For instance, if my partner generally wasn't much of one for attaching meaning to pet names, I wouldn't take it personally. If this had always been the status-quo, I wouldn't take it personally. If, however, my partner took a name she had only ever called me (i.e. it has meaning), and then applied this to someone else, I'd find that a shame.
What if your metamour started calling your love by the same pet name you have for him, would you find a new name?
That's an interesting question. If the name had a lot of meaning and was quite specific, I'd actually find that intrusive. Honestly? I probably would end up losing enjoyment of using the original name I had chosen.
I should say though - I'm a writer, so words carry a lot of weight and meaning to me. Arguably too damn much.

Yes, there have been times that my GF has used terms with her other partners that I have found meaningful between us. I tend to agree with Marcus, though: we can't possess words or terms of affection. I try to let it go when this happens.