I know there's been a lot of back and forth on what counts as a service animal vs assistance animal, laws in the US vs Australia, but as pretty much all RCI ships are flagged as being registered in the Bahamas technically Bahamian law would apply.
(tl;dr summary of below--not only are some breeds outright prohibited even as service animals, "service animal" is defined strictly enough under Bahamian law that "self-trained" dogs don't qualify unless they're explicitly trained to perform an alert or "protect" action, and therapy dogs and "emotional support" animals certainly don't qualify.)
For starters (unlike many places in the US, but similarly to certain provinces of Canada as well as the UK and much of Europe) there are "breed ban" laws that essentially ban "pit bull" type dogs and crossbreeds (including not only the four breeds generally considered as true pit bulls, but Cane Corso and Dogo Argentino derived bullmastiff breeds)--and no, you cannot bring a "pittie" on board or into the Bahamas, not even if it has been fully certified as being trained to perform a service such as a "Helping Hand" dog or seizure-alert dog or PTSD alert dog (probably the three most common categories you'd see "pitties" performing legit service animal duty). And yes, there do seem to be a fair number of "pitties" trained specificially as PTSD dogs for veterans in particular, so that's an important thing to keep in mind.
Bahamian law also explicitly defines an assistance dog as "any dog that is individually trained to do work or perform tasks for the benefit of an individual with disabilities, including physical, sensory, psychiatric, intellectual, or other mental disabilities" (Animal Protection and Control Act, 2010); "self trained" animals are not legally recognised as assistance animals in the Bahamas unless they have had some form of formal assistance dog training. (In the case of legitimate PTSD alert dogs, this would be specific training to sense when the owner is in fact having a PTSD crisis and performing a specific action--covering their human, or getting their human away from a stressor. For dogs sensing seizures, a dog's trained service would be to explicitly get their human to a safe spot and remain with their human until the seizure has passed. In the case of "hearing ear" dogs, the trained service is to alert the owner to sounds in the environment using visual cues. For "glucose monitor" dogs the trained service is, again, to get the owner to safety and to alert the owner appropriately if it smells the blood sugar is going too low or too high.)
"Emotional support animals" not trained to do a specific alerting service or specific assistance task do not legally qualify as assistance animals, and technically "Delta Dogs"/"Pet Partners" legally would not qualify as legitimate assistance dogs under Bahamian law (or even under the US's ADA) as they are not trained to perform a specific task including alerting or getting a human to safety. (In fact, Pet Partners--the American affiliate of the Delta Dogs programme--has actually issued a formal position statement explicitly stating "therapy dogs are not assistance animals" and therapy animals should not be passed off as assistance animals.) as
Furthermore, again under Bahamian law (section 28 of the Animal Protection and Control Act, 2010) legitimate service dogs under Bahamian law (again, that are explicitly trained to perform an essential service including alerting of owner to health conditions) are still required to be controlled by their owners, and "the person whom the dog is accompanying must comply with any reasonable conditions imposed by the occupier or person controlling the premises or place in relation to the entry or presence of the dog". This includes prohibition on animals being "at large" as defined in section 10 in the Act, which includes prohibition on the dog being a "public nuisance". Bahamian law further (in section 27, subsections 1(b) and (c) of the Act) explicitly list as offenses any act where a dog "deposits faeces on property other than property of the keeper and the person fails to collect the faeces and dispose of it in a reasonable manner; or repeatedly soils, with urine or faeces, property of, or under the care of, another person without that other person's consent". Section 27, subsection 4 of the Act further explicitly notes anyone taking "a dog into a public place in contravention of subsection (1) commits an offence".
So if anything, the laws in the Bahamas (which is the flag registry of RCI's fleet) are actually rather stricter than in the US law (US law allows service horses for persons allergic to dogs but requiring a service animal, among other things).
In addition, the Bahamian government (unlike the US) actually has explicitly authorised a certification scheme for legitimate service animals (specifically in section 23 of its equivalent to the ADA, the Persons With Disabilities (Equal Opportunities) Act, 2014) to be done in conjunction with the Ministry for control and protection of animals. (In other words, unlike the US, yes, the Bahamian government CAN very much ask "What service does this animal provide?" and even ask for specific paperwork from the training facility or (for personally trained animals) documentation of the training programme and the specific task or tasks the dog performs.)