I just discovered the exact same issue - I have an 1100 sq ft basement with zero visible water issues - we're down there all the time. Been here a year through rain downpours, etc. and not had any problems. But we had a foam rubber exercise mat down and when we picked it up, there was a little sheen underneath - it was moisture. There isn't any elsewhere and now that we're wanting to put tile in we're not sure if it's condensation or seepage. I did a test and moved the foam mat to a different location - the old location dried up, but the new location developed a sheen underneath in a matter of a few hours. We're not sure if we should proceed with tiling, put down a vapor barrier or treatment first, or stop tiling altogether and make some major repairs.
Yes water vapor can get under the tiles f the tiles are not completely non-porous.
Try putting a sheet of plastic (at least 4x4 feet) down on the floor and put some of the tiles or exercise flooring on top of the plastic. See if moisture appears under or on top of the plastic.
If the ground temperature is lower than the room temperature and the humidity in the room is not particularly low, then there is a good chance of water passing through porous floor material and condensing on the concrete floor underneath.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.