OSMBuildings | 3d building geometry viewer based on OpenStreetMap data | Map library

For use with Cesium, you probably want Cesium 3D Tiles. There's a Cesium 3D tiles demo based on OSM building data for New York City which proves that it's possible, but that looks like a one-off with limited geographic coverage.

I'm not sure what OSMBuildings is currently using, but it does not seem to be the same as Cesium's 3D Tiles. Some older info on GitHub mentions using GeoJSON, but looking at the network traffic, it now seems to be using Mapbox Vector tiles, which is not a format specialized for 3D data, but rather a general-purpose solution for transmitting OSM data (and other data sources) as tiled vector maps. On osmbuildings.org/data, they mention that they are willing to provide data in other formats for commercial customers, though, if that's an option for you.

Finally, some people have experimented with providing OSM for Cesium using open source software (see e.g. the osm-cesium-3d-tiles and osm2cesium repos). This might be a starting point for setting up your own service if you're willing to go down that path, but it's definitely not a complete and polished solution at this point.

To sum this up, OSM has the data that's necessary to set up such a service, but the various providers use different formats as there's not really a clear standard yet (unlike with 2D raster and, arguably, vector maps). So to the best of my knowledge, there's no ready-made service in your desired format available today.

Looking at OSM Buildings examples the "YOUR_MAPBOX_KEY" is used when making tile requests to MapBox servers, like:

The height is contained in the JSON file.

See this example here: https://b.data.osmbuildings.org/0.2/anonymous/tile/16/35210/21491.json. It contains a properties field which in turn contains height and levels:

If your screenshot is up to date, you're currently styling your building_custom layer as a fill layer. In order to make them 3D you'll need to change this to a fill-extrusion layer (next to "Type" — the fill-extrusion layer type is the one that looks like a windmill). Once it's a fill-extrusion layer you'll be able to adjust its height in the Style panel.

As to your other question I think what you're asking is why building_custom has a light purple line next to its name? If this is your question, the color coding has to do with what source-layer a layer's data comes from. The default building layer comes from the "building" source layer in Mapbox Streets; your custom building layer is from a different source layer. (All the place-* layers come from the same source-layer — "place-label" in Mapbox Streets — which is why they all have the same color.)

Solution: shim leaflet itself, and let it export 'L'. Then putting the deps in plugins will work. My full config: