Ooooh! One of my favorite topics... Piggy!
So, yeah, I'm with Ru and Heralde here, quelle suprise. Piggy is definately quite isolated, and Heralde is right too, in that at least some of it, Piggy does to herself.
However, there are almost always reasons for why we do what we do, and why we are the way we are.
In my job, I work a lot with children who remind me SO much of Piggy. She's got such a 'Veni, vedi, vici' attitude towards life. ('I came, I saw, I conquered') Everything is a battle... Piggy stragizes, and plans and works and fights for anything her heart desires, and she's very nearly incapable of taking no for an answer, let alone taking no for an answer gracefully. It makes her tough to handle for anyone trying to work with her, or heaven help them, give her orders she doesn't see them reason for, or finds slighting.
She's always watching for an attack, either professionally or personally, and she's got no patience for even affectionate teasing because of that. Often times, we veil truth in a joke, and Piggy is painfully aware of that.
This is probably half nature, half nurture... maybe more nurture. The kids I see this attitude in have very tough lives... They do not believe anyone is out to help them without an ulterior motive. Once they trust someone, they'll do anything, anything to back them up, even if that person is in the wrong. If they think you're snubbing them, they don't use reason, they fight back... they stand up for themselves.
Sound familiar? Piggy is not someone new to fighting... she's a PIG in show business...and show business is brutal like no other. I can't imagine it was kind to PIggy in the beginning, not at all. I seriously doubt there was anyone to believe in Piggy's career before she met Kermit but Piggy. She's her number one supporter in a world that mocks her dreams... and she's not about to tolerate that attitude from anyone, least of all the people who call her friend.
So that's her career... now for her frog... well, that point has been made beautifully by others. Same principle applies anyway... a little competitiveness between friends is normal. Happens all the time... has nothing to do with any sort of romantic feelings between Kermit and Fozzie!
If anything, I think Fozzie's jealousy is almost a response to Piggy's overt possessiveness... and she's got a lot more to worry about regarding the possibility of losing Kermit. He's not remotely indecisive or evasive about his friendship with Fozzie after all.
I think Piggy has certainly warmed up to the other muppets... she counts Beauregard as a friend, oddly enough, and she does love them all... of course admitting that would leave her vulnerable, and hasn't done her much good where Kermit is concerned. He's a kindhearted frog, but he has left her high and dry more than once, exposing that chink in her armour for the whole world to see and laugh at. Hard to imagine someone already as much on the defensive as Piggy is to let herself be emotionally vulnerable to more of them... and that dropping of defenses is a huge part of feeling close to someone.
So... yes, Piggy is not so friendly and trusting of the other muppets as Fozzie and Kermit are, and she plays quite a large part in keeping the others distant with her behavior and her over the top personality. But it doesn't come from the void, and she's not the only one at fault.