I'm not aware of any place that specifically has the word Caribbean in its name in Long Branch. Viking mentioned a couple places below, there's also Mark's Place in Asbury and Blue Marlin in Bradley.
There's a restaurant in Asbury that is completely unknown to the world (no Yelp page even) that has been there well over a decade. I think it's literally called "Caribbean Restaurant." But I'm pretty sure the people who run it are Peruvian. They have that green sauce aji.
I can pretty confidently say that the best oxtails in our area are at the Cuban place in Asbury and Jameson's (Soul food, not Caribbean) in Neptune. The only place I'm aware that has oxtail that I haven't tried is At the table because they've always been out.
Otherwise I'm reasonably certain that no local Caribbean place is better than any other in any significant way. Personal preference and luck of the day will be the biggest factors.
The following are the places I am aware of that are Caribbean/West Indian between Asbury and Long Branch:
The Islander II
There are also 2 Haitian places. One on Asbury Ave, and the other on Memorial. The place on Memorial is always deserted and really unapproachable. They don't even have a menu. You walk in and whoever is "working" just looks at you like some kind of weirdo. LOL. I've gotten a few things from there but nothing made me really want to return. I think Haitian food in our area is just not very exciting from my limited exposure. They seem to just fry everything until it's a rock. (I don't know anything about Haitian food outside what I've had here)
The other place on Asbury has an incredibly inexpensive buffet but nothing there will really wow you.
In Long Branch I think the only Carib option is Norah's.