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For example on wikipedia for Switzerland it says the country has an area of 41,285 km². Does this take into account that a lot of that area is actually angled at a steep inclination, thus the actual surface area is in effect larger than what you would expect when looking onto a map in satellite view?
No. It's a flat approximation. The short answer is that once you take account for topography, your answer will always grow with surface resolution, and thus the actual surface area of rough topography is undefined.
It's the same problem with defining coastlines. You can keep increasing the resolution and the coastline length will increase indefinitely.
And what about navigation, does it count the slopes in? Is the route actually longer than it says if you travel up and down mountains?
It's probably aware of them, but generally no. Most slopes for driving on are smooth enough to be pretty negligible unless you're going hundreds of miles or more, in which case fives of miles won't make much difference either.
But if you're traveling by bike those small slopes may make some parts of the ride significantly more difficult or easier, and for cars may impact fuel efficiency in a way much more significant than just counting the extra distance traveled. So many navigation systems will still account for slope, even if they don't necessarily acknowledge the length of your path as precisely as you may have hoped
I can't remember if it was Matt Parker or Tom Scott who did a thing about this on youtube, and I think the answer was no, they just use whatever method is easiest to get these numbers, and they're not even using a consistent methodology from one country to the next.
Inclination must be ignored for area to work on a map, plus the inclination ends up being lost in the noise on a large scale. It is very similar to the coastline issue where the more detail you include the longer the coastline gets until every coastline is basically infinite.
Let's take an area split into a grid. One are has a hilly round slope, one is flat, and the rest are a variety of combinations. If you tried to take slope into account the one with the round hill would require the straight lines of the grid to warp towards it like one of those space time curvature pictures. The one that is flat is the only one that could be square, and even then it only works if you count it as flat since even flat ground has a small texture.
So no, they don't take elevation into account for maps because it would be far too difficult to measure.
The surface area can only be calculated with a defined level of accuracy due to how textured surfaces work.