Kind of. There certainly are times when they vary the approaches. But my experience is that even when they switch runways they still come in from the south over brooklyn with manhattan on the left, then do a loop over hell gate and the bronx and come back in. So either way it's on the left.

Like anything there are exceptions but as far as I can tell the active runway changing usually doesn't change that larger traffic pattern. Your view may vary.

EDIT - This made me curious so I looked up the STAR (Standard Terminal Arrival) sheets from the FAA for LGA. Not a single one of the standard approaches show an arrival with Manhattan on the right. Most on the left, and a couple mostly straight ahead but slightly better on the left, coming in over the sound.

The ILS approaches on the other hand look to be from the North, South, or West, the latter two of which would be bad for a left-seated passenger. But then again that's likely to be in non-visual conditions so it wouldn't matter.

So it seems like "manhattan on the left" is indeed the standard. Though any real ATP pilots that want to make me look foolish for misreading the approach data should chime in now.