It’s at least a Soft Oligarchy yes. There’s no legally or extra legally enforced class system. If you can make it into the upper classes, by guile, luck, or sheer bastardry they’ll accept you and let you run some things, maybe even political offices. See J.D. Vance, a millennial from Appalachia who has risen to the Vice Presidency via guile and sheer bastardry.

But it’s a Soft Oligarchy because opportunities are far from equal. Before anyone starts screeching, equality of outcome isn’t an expectation here, merely equality of opportunity. In the large majority of cases your zip code can predict your future socioeconomic level. And not because rural areas are cheaper, that just means middle and upper class start at lower numbers there. Those classes are still not being obtained. Along with this are several studies over the last couple decades telling us that socioeconomic mobility is all but dead, both individually, and more recently, intergenerational mobility.

So while you aren’t going to be killed or imprisoned for earning too much or asking for stuff above your station, it is very rare to access those levels without being born to them. Thus the “soft” in Soft Oligarchy.