Most likely it has less than 2 GB of VRAM. That the blue screen happens at 2 GB doesn't necessarily mean anything. It's mot likely that the blue screen happens when it's really starved for bandwidth; GPUs, even junky ones, will just use slower system RAM when it runs out, so it's unlikely to see a blue screen right at the GPU's VRAM limit.
This is a weird one. The PCB looks an awful lot like a 550 Ti to me.
But it's weird because of the chip.
You have what looks a lot like a GTX 560M; it has all the same identifiers as a known 560M. And there's no particular reason why someone would want to make this
look like a 560M. But that shouldn't be in a desktop GPU.
So it's a very, very odd situation. It makes me wonder if there were some poorly documented and obscure China-only chips out there for the 500 series. I'd bank on the chip being from this era, but I'll be damned if I know which one.