Will using a paint or sealer on a concrete floor damage it?
Once concrete is fully cured, you're not going to damage it with paint or other sealer.  On the first floor of a building chances are good that it is a solid concrete floor rather than a thin coat of something.  As far as the "type of concrete", that shouldn't matter much when you choose a coating.

What does matter is the prep work that you do.  When you decide on a coating, read what the manufacturer recommends.  In most cases it will call for a strong cleaner/degreaser and some type of concrete "etching".  Etching might sound like it could damage the floor, but we are talking a very mild etching that happens on an almost microscopic level.  Don't let the term scare you.

The amount of prep you do is directly related to how long the coating will last.  Skimp on the prep, and it will start to peel and flake off prematurely.  It's not going to hurt the concrete, but it could make the floor look very ugly.