No, OSM does not contain multiple layers in the GIS sense, and cannot, at least not without extensive modification AFAIK.
OSM does have a layer tag (https://wiki.openstreetmap.org/wiki/Key:layer) but that expresses an entirely different concept, that of the vertical relative position of features, for example a bridge is higher level than the river that it crosses.
There are no “declared” layers in OSM there are only nodes, ways, and relations, each of which may have tags. However, if you are using an extract from OSM, such as the shapefiles from GeoFabrik, you will see that the process that created the extract has translated the OSM data into GIS layers (but not thousands of them). However, that is just so the data can be used in a traditional GIS, such as QGIS. A different provider of extracts from OSM may choose to do the translation differently.
Highly unlikely in any spatial database, let alone OSM. I can only think of a couple of dozen polygons that a given point may be in. Nation, state/province, county/parish, city, some political subdivision of a city maybe, one or two land use polygons, a building, a body of water (but probably not at the same time as being in a building), park district, fire protection district, National Forest boundary, military base, watershed, a special taxing district or two. I am sure I have omitted some, but there are that many, and some of those may not be mapped in OSM.