No, PS1 games on PS2 were running like on the PS1 as the PS2 had PS1 components, same for the first models of the PS3 with PS2, and PS1 was emulation on PS3. But no improvement, like Virtual console.
PS1 software was still very timing dependant so Sony avoided the changes there* - not because of some hw reason (the heavy lifting of PS1 BC was done through emulation on PS2), but because it guaranteed compatibility. Performance 'did' change somewhat from 7xxxx series onwards though (which added a few more titles with compatibility issues to the list).
They also offered fast-load times and filtering - but both options would break in certain games, so it was off by default, and user enabled only.
Future PS1 emulators would continue this trend - PS1 on PSP, Vita and PS3 alike offered fast-loads, but still as a toggle-option. I haven't really tried it since the PS4 so can't say what's it like now.
*There are select PS1 titles (very few were unlocked like this) that run better (Quake 2 and Forsaken come to mind) - both had a 60hz target that original hardware missed relatively often, but it is comparatively stable on Sony hw from PS2 and onwards.
PS2 compat - after switching to emulation - offered 'usually' improved performance (PS3 wasn't always faster, but usually, and since PS4 framerates are bumped by 2-3x on unlocked games).
But also worth noting unlocked fps and adaptive-Vsync was still comparatively rare in that era, for first few years Sony did not allow it at all on the platform.