For those who are curious, the building age data comes from the Dutch government (as others already said). It was imported into OpenStreetMap about 5 years ago by a few dozen volunteers including myself.
The data contains addresses and buildings (outline, building started date, and an identifier to make updates easier). This means we removed all existing buildings and addresses in the country, which felt really risky, but because the import was done manually and only tool-assisted, it seems we managed to get it right. Took a few months, but once enough volunteers hopped on, it went quite fast. If I remember correctly, once you got the hang of it (your third import or so), an average municipality took about 2 hours (we have 350 municipalities).
I am not sure where Mapbox' height data comes from. In OSM, you can give a "levels" tag to indicate how many levels a building has (from which the height can be estimated), or a height tag in meters if it is known, but the buildings don't have that. Perhaps it is just assumed that large buildings are a bit taller and building=house is an average house height unless otherwise noted? Or maybe they use the Dutch AHN, which is a very accurate height scan of the Netherlands. Some people made it into a 3d picture, e.g. https://snipboard.io/Sq6C38.jpg (this only uses AHN data, not OSM, so it only has the tops of objects as points, but that's still quite an accurate outline).
To be more precise: it was imported en masse once, and has been meticulously maintained via localised imports ever since. OpenStreetMap in the Netherlands shows how open data can flourish when volunteers and government collaborate.
The data contains addresses and buildings (outline, building started date, and an identifier to make updates easier). This means we removed all existing buildings and addresses in the country, which felt really risky, but because the import was done manually and only tool-assisted, it seems we managed to get it right. Took a few months, but once enough volunteers hopped on, it went quite fast. If I remember correctly, once you got the hang of it (your third import or so), an average municipality took about 2 hours (we have 350 municipalities).
For the Dutch speakers among us, more info about the project can be found here: https://wiki.openstreetmap.org/wiki/NL:BAG
I am not sure where Mapbox' height data comes from. In OSM, you can give a "levels" tag to indicate how many levels a building has (from which the height can be estimated), or a height tag in meters if it is known, but the buildings don't have that. Perhaps it is just assumed that large buildings are a bit taller and building=house is an average house height unless otherwise noted? Or maybe they use the Dutch AHN, which is a very accurate height scan of the Netherlands. Some people made it into a 3d picture, e.g. https://snipboard.io/Sq6C38.jpg (this only uses AHN data, not OSM, so it only has the tops of objects as points, but that's still quite an accurate outline).