In pinnipeds I think Europe is ahead, but that obviously depends on how you rate the individual species. Harbour is the standard seal in both continents and California the standard sea lion; both very numerous in Europe and North America. Steller's sea lion and northern fur seal far less numerous but still in a good handful of places in both continents. Grey seal common in Europe; only somewhat less so in North America. Harp seal is present in tiny numbers in both continents (less than a handful in total). Walrus, present in relatively small numbers in both continents, has been discussed in detail in
this thread.
Pinniped species in Europe (* = Russian Europe only), but as far as I know not in North America: South American sea lion, South American fur seal, brown/Cape fur seal (fairly common in Europe, not sure if the single North American at Lousville still is alive?), ringed seal (any in North America? Assiniboine?), Baikal seal*, bearded seal, ribbon seal*.
The only pinniped species that are in North America, but not Europe are Hawaiian monk seal (Minnesota; also Waikiki - politically but not geographically North America) and Guadalupe fur seal (SeaWorld San Diego). Not sure about the current status of
northern elephant seal in North America. There might be a few, essentially occasional rescues.
Northeast Pacific fish and inverts are far better covered in North American aquariums than European, but that's the same issue as other native species discussed earlier. Similarly, northeast Atlantic fish and inverts are far better covered in Europe than North America. Marine fish and inverts from other regions are overall fairly equal in coverage in North America and Europe, but North America does have three "big ones" not present in Europe: Whale shark (as mentioned by Jurek7; Georgia Aquarium), manta ray (Georgia) and large tunas (Pacfic bluefin at Monterey Bay). Freshwater fish and inverts follow a similar pattern: Unsurprisingly native North American far better covered in North America and native European far better covered in Europe, with species from other continents overall equal.