It's basically something they deliberately play with in productions (where continuity varies, since they *are* movies made by the Muppets), and even moreso in interviews and elsewhere. See Are Kermit the Frog and Miss Piggy married? where we try to cover all m,arriage attempts and all discusses of their relationship status.
Note especially the quote by Jim Henson about Muppets Take Manhattan and the casting of an actual clergyman in the part, and in pieces on the making of the movie and Frank Oz and so forth, it's also brought up as a deliberate ploy for ambiguity: "Kermit says - and will continue to say - on interviews and such, "I'm just an actor and when two actors marry on stage, they're only acting." But Miss Piggy continues to bring in this minister as evidence that they're really married. So, the argument will continue on hopefully into — I don't know what — we'll wait and see."
In The Muppets (2011), they chose to treat it as continuity (the photo from MTM). In most interviews, they've gone more recently from Kermit denying all involvement to admitting a relationship but denying marriage status. In the most recent (it's missing from that page, but I *think* it was in the Best in TV special), when pressed, Kermit says he and Piggy plan to legalize their relationship only when all species can marry each other.
I really don't get why they've shifted from the standard "working relationship" (he thinks they're working; she thinks they have a relationship) to this new, pointless "yes we're happy together" stuff. It dilutes one of the major driving forces of each of their personalities.
I think that approach doesn't give them anything to make a movie about. The romance between Kermit and Piggy was a major plotline in the last movie -- just as important as Walter's storyline, and Gary's relationship with Mary. For that to work as a real story, they needed to decide -- how do these characters actually feel about each other? What changes over the course of the movie so that it feels like a story?
The "working relationship" line is funny, and they've been doing variations on that since 1984 -- which is the moment that they stopped telling stories about Kermit and Piggy. They only talked about it in guest spots and talk show appearances, and the movie storylines had to be carried by the human stars, or Gonzo and Rizzo. I know what you mean about diluting their personalities -- but I'm happy to see them letting Kermit and Piggy drive a story again.
Plus, even before Jim died, although again mostly in the guest spot area, they occasionally had Kermit go out on a limb to show real romantic interest, see especially his appearance on Reading Rainbow when LeVar Burton finds him in the pig book section, and then with a copy of How to Pamper Your Piglet (but he was quick to conceal it). I keep meaning to transcribe that whole scene.
The problem is I've never found the prospect of a relationship here compelling. They have virtually nothing in common. Piggy's only words of explanation for her attraction tend to involve Kermit's eyes or other physical attributes, while I'm not aware of any attempt to explain what Kermit sees in Piggy. Well, there were the meet-cute scenes in tMM and tGMC, I suppose, but those always felt forced too.
Maybe I'm just a fuddy-duddy. I miss the days when Kermit was trying to avoid Piggy's advances.