The picture you linked, having 40cm of clearance, and being typical of offices sounds like you are describing a 'drop ceiling' (aka 'suspended ceiling') -- this is different than drywall.
A drop ceiling is composed of 2x4' or 2x2' removable panels sitting in a metal grid system, and the panels are usually mineral fiber or fiberglass but can also be wood or plastic.
You absolutely can remove this, but there are of course drawbacks
It will definitely heavily affect acoustics in the basement. In fact these are also sometimes called 'acoustic tiles' because they do absorb sound. By contrast a concrete ceiling is going to make the room very echoey.
As far as separating noise to the upstairs, I suspect it will not have a huge effect, though it'll definitely be non-zero. Concrete does transmit low frequency sounds, and a soundproofing layer can help to minimize this.
The best way is to lift a tile and inspect the concrete. Does it look decent enough for your tastes? Is it full of holes drilled to hang the ceiling, or does it have wiring or other utilities running across it? If there is any wiring, plumbing, HVAC or gas lines installed by people after (or knowing about) the drop ceiling, they may not have done as neat a job as if it would be exposed.
Replacing with drywall
An actual drywall ceiling can look pretty nice. For reference/comparison, that would be something like this (picture is before painting):
One of the benefits of a drop ceiling is after it's installed, you can still easily run wiring above it -- for example if you want to install a home theater, or upgrade to the next generation video cables, or go to a 7.1 or 9.1 or 24 channel surround system.
Drywall requires cutting holes, patching and repainting -- or having strategically run conduit behind it to exactly the places you need.
Other drop ceiling tiles
There are ways to install low-clearance drop ceilings. The minimal you can get down to is around 2 to 3" (5 to 8cm) -- it just depends on being able to get the tiles in.
You can also replace tiles. There's some neat tiles that definitely don't look like your typical office -- just be aware some may lose some of the acoustic benefits. You can compensate for this by insulating above, but then you need more space.
Finally, you can just paint the tiles, which is easy and cheap and can vary the look quite a bit.