If you turn off the 2 pole breaker in the electrical panel the SPD (Surge Protection Device) is connected to, the SPD will be disconnected, isolated, from the electrical panel.
That is not how this type of surge protector is connected, if it is what I'm thinking of, like this model:
https://www.alliedelec.com/square-d-qo2175sb/70060775/?mkwid=sUgs25dac&pcrid=30980760979&pkw...
It's impossible to tell from the pic, but the surge protector itself takes 2 spots. In this way it is just like a 2-pole breaker. It connects directly to both phases with no interceding breakers. There is a tail to be attached to the panel's neutral.
In short, there is no breaker on the protector, and no breaker that is assigned to power the protector either. Should the surge protector itself short the panel's main breaker will have to trip.
There's no way for a homeowner to "turnt it off" short of removing the panel cover and pulling it off the panel.
Best,
E