There are 326 bases in the US. https://militarybase.net/u-s-military-bases/ (see the CSV list at the bottom). The US has a total of 50 states and 6 protectorates (Guam, Puerto Rico, etc...). There's no where near 80 bases per state unless you count National Guard Armories (parking lots with a gym in most cases) and recruiting offices. The US has a big military, yes, but not that big.
In retrospect, you are a bit right here. Pardon the bike shedding, but maybe that is not a precise definition due to the vagaries of what "military", "base" and "installation" means. Does room 641a count? I'm not sure. The US Military has its own very special definition, but the general public would know its own.
Under this expanded rubric, National Guard, Coast Guard and Reservist armories would definitely count, CIA black sites[0] would certainly qualify as well (and they are not in your list to be sure), in addition to listening posts, air fields, possibly also foreign naval supply points if they meet some of those continually manned criteria (and hell, why not? Its in the spirit of this thing, we'll just consider them 3/5ths an installation for the purposes of statistics), some forward operating bases if they are continually manned and resupplied for, lets just hand-wave 11 months. Also, private contractors are definitely in this if they have their own separate facilities and especially if they conduct military operations such as "security forces" in Iraq. Now that I'm thinking about it, air craft carriers should be included as well if they are on deployment ... I know, I know, thats pretty rare.
But you are right if you point out that this isn't a precise measure, that it changes continually, and that I don't actually know how many there are--it could easily be tens of thousands! Just scale the definition until it feels most comfortable for you.
That is the real point, anyhow.
[0] - https://prisonphotography.org/2010/03/30/extraordinary-rendi...
"only" was used because the op said "thousands".
There are only 750-800 US military installations around the world. Most of these are small. This number has been reduced by about 1000 since the peak in the 1980s (cold war era). The majority of large installations are in the US.