OSM doesn’t use an API to “show a location on a map”. The openstreetmap.org site uses a JS library called leaflet to do that. A URL such as https://www.openstreetmap.org/#map=6/18.041/80.079 encodes the zoom level, latitude and longitude. The underlying map tiles can be whatever you want, and you can host your own (for example https://map.atownsend.org.uk/maps/map/map.html#zoom=13&lat=52.833&lon=-1.2667 is a simple map view incorporating tiles I’ve created). See also e.g. https://openmaptiles.org/styles/ for vector rather than raster tiles.
There are external APIs for forward and reverse geolocation, of which Nominatim is one (see https://wiki.openstreetmap.org/wiki/Nominatim ) but unlike with Google there are several instances around the world, and you can host the data yourself for free.
I’m not aware of a drop-in replacement for Google Map API calls to replace them with OSM ones (my recollection was that one did exist for Cordova or Qt, but I can’t find a link right now).