Yes, everything that has been published in the PSN store under the label "PS2 classic" runs using the PS2 emulator named ps2_netemu.self
If the game was published in the PSN store it means it passed the quality control and is good enought to be sold (in other words, the game should not have any problem), otherway they could not publish it

Imagine you pay for a PS2 classic in the PSN store but the game have problems with textures or glitches... this is not aceptable
The detail you should keep in mind is some of that PS2 classics published in the PSN store contains a CONFIG file that allows either to configure "settings" of the emulator or to "patch" the ISO/RAM on the fly
If the game had a CONFIG for it then we should consider that CONFIG is perfect and you should use it because is a official CONFIG made by sony
The other PS2 games not published in the PSN store, some works fine without CONFIG, others have a non-official CONFIG that fixes the problems, and others doesnt have any CONFIG because the attempts to make a CONFIG specific for them was not good enought