You can pretty much go to sea in an uncertified bath tub if you wish, and it's not clear that a flag or coastal state can interfere.
The Canadian mother ship has to meet IMO requirements, including things like SOLAS (the safety at life at sea treaty) and MARPOL (the marine pollution treaty).
The submarine is not required to comply with any legislatory requirements when it's outside territorial waters, due to its size, and the fact it is not undertaking international voyages under its own power, and it's carrying fewer than 12 passengers.
For manned submersibles, it is possible to get them classed by an international classification society, like Lloyd's Register. They have, or certainly used to have, rules for the construction of manned submersibles. These rules cover the pressure hull, critical electrical systems, stability and the like.
For things like the pressure hull, it is possible to have it certified under a pressure vessel code, such as those published by ASME. That would determine the safety factors for the design and test pressures required for the main hull, but would exclude everything else. From the press reports, the plexiglass dome sounds like it was certified for a lesser depth than the operating depth of the submersible.
As you say, millionaires tend to be very litigious cargo. I am sure there will be a lot of lawyers looking at this, and there may be insurers involved, but I would imagine the public liability side of things will be, can we say...watertight.