The problem is, the mass produced chicken most of us are used to doesn't taste like chicken.
The expression 'tastes like chicken' is therefore used for most anything that's not particularly assertive.
There was an episode of Food Detectives, "Tastes Like Chicken" where they served a few different meats to professional chefs (Alex Guarnaschelli and Aarón Sanchez), who had problems identifying all of the meats they were being served ... I can't remember exactly what all of the results were, and I can't find a transcript; I remember Alex being distraught when she realized she was eating guinea pig).
And the cooking show Ham on the Street had a regular segment of 'name that meat', where he'd have hot dogs, or jerky, and try to have people guess what they were.
Part of the issue with substitution is going to be something that has similar cooking characteristics, but that you don't have a reaction to. (I'm actually surprised that you're allergic to chicken, but then listed other poultry)
Turkey's likely the closest of the stuff you listed, particularly if it's the mass produced stuff, it's going to have a similar lack of flavor that most chickens sold in the US has; duck tends to be richer. I haven't had any game birds recently, so can't comment on the pheasant. Alligator I've only had a few times, and the only time in the last decade it was deep fried, so the meat didn't come through.
For substitutions, it'd depend on how you're cooking it, and what part of the bird.
For instance, in place of chicken wings, I might consider frog legs. (although, it's probably been 25 years since I've had 'em, but I seem to recall them being chicken-like ... I just can't remember if they were like white or dark meat chicken) ... but they'd be closer in mass than turkey wings.
For chicken breasts, I'd go with turkey breast cutlets, so they're closer in size. For ground chicken meat, ground turkey will work, just check to see if a recipe calls for a leaner or fattier mix, and get the appropriate turkey. Chicken thighs and legs are a little more strongly flavored ... rabbit might actually work, or duck or pigeon if you have a source.