Many countries do, but I wouldn't vouch for all countries. It's a reasonable starting point for whatever country you live in, but probably not the easiest starting point if you're looking for data on countries where you don't speak the local language.

In the US, at least some states also maintain a set of state-level databases, and I've even seen county-wide and city-wide databases -- though less frequently.

There's also a lot of sharing that goes on (the Canadian CDSM database started with NASA/JPLs SRTM database), and there's a lot of variation in available resolutions, too. For example about 70% of England has been scanned with LIDAR, but last time I looked, California has very spotty coverage -- predominantly corresponding to wildfire locations.