I wouldn't have an issue with getting a certification from any of the well known certification agencies. I have certs from 3 different agencies (PADI, YMCA, SSI), and could have had two more (NAUI & CMAS) as they were offered for an additional fee during the YMCA course. However, since I already had PADI OW, I didn't need the others for acceptance concerns.
I would be hesitant of some certification that I had never heard of. I would at least do some research into them to see if they would be recognized or not.
There have been shops that have issued fake certifications in the past. One in S Florida was issuing fake PADI certs before being busted. That's easy enough to research. First, look at the shop to understand what agency(ies) they certify to. Then, go to the agency's website and make sure the dive shop/instructor is listed. If so, all good. In the case of the above, searching PADI's website would have shown no such affiliation.
The instructor is more important than the agency.
One thing to note, though, is that the certifications may carry the same name at different agencies, but aren't really the same thing. That could lead to some confusion. Advanced Open Water/Scuba Diver for example has different meanings at some organizations.
For PADI: AOW is an overview into some specialties. It does not equate to a full specialty.
For SSI: AOW is a recognition of completing 4 separate specialties.
For SDI: ASD is a recognition of completing 4 separate specialties.
For NAUI. ASD looks like an overview, but I'm not positive.
Normally, this wouldn't be an issue. However, some charters may require certain certifications for certain dives. Deep and Night/Limited Vis for example. In the case of PADI and NAUI, even if you did Deep and Night/Limited vis dives as part of the AOW cert, it's not a full specialty, so might not meet the requirements. This would all be dependent on the dive charter and their requirements.
Both SSI and SDI offer Advanced Adventure course which is closest equivalent to PADI AOW. SDI also has this equivalency chart on their site.
Recreational Sport Diver Equivalences - SDI | TDI | ERDI | PFI