I think the potential issue with supermarket/ bank/ other insurance is that it's not their core business. So in the event of a change in the market or it not being profitable, they may choose to withdraw the service. In which case you may be left with having to shift insurer with all the attendant issues if your pet has been treated for any conditions in the past. Think this happened with Lloyds?
With specialist insurance firms like Petplan, one would hope they are more likely to stay solvent (although no guarantees).
Having said that, Allianz underwrites both Sainsbury's and Petplan and I've had no big issues with either - 2 dogs on Petplan, one on Sainsbury's. (Sainsbury's changed underwiter when AXA withdrew from the pet insurance market - so it doesn't mean all supermarket companies are necessarily quick to withdraw their products when things change).
I think, though that many of the supermarket policies are often offered at very cheap rates upon first subscription, and seem so much cheaper than say Petplan. However a few years on, they are often on a par with, if not more expensive than specialist companies as the initial low subs are designed to capture new clients.
Probably doesn't help.