Repairing parquet: Is it wood?

Oops.....regarding your question about sanding by hand or machine: You don’t have to sand away any of the wood itself, but only remove or roughen up the existing varnish or similar, so the new material adheres better. Therefore, you could also sand a small room by hand and rather lengthwise, that is, along the grain. Of course, you can also use, for example, an orbital sander or similar, and I would say not to use paper rougher than 120 grit, maybe even somewhat finer. Depending on which room (children’s room, study, or similar), I could also imagine using the [Dekorwachs deckend] straight away, there are really nice colors. In Scandinavia, they have it in grey, white, light green, coffee.....like I said, depending on the room. Meanwhile, Osmo has renamed some products (older products are still available online), mine was called "Dekowachs deckend" back then, but similar products surely exist from other companies as well. What I liked was that I didn’t have to sand and the result. Again though, this is my personal opinion and how I would do it based on my own experience, all other statements here are equally valid.

I can only agree with pagoni2020. Sand along the grain. In my opinion, use a handheld belt sander. Start with 120 grit and then do a test with the sealant; if sanding marks are still visible, sand again with finer grit (250).

The thing with decorative wax also sounds interesting. So I could, for example, brush on decorative wax "oak". Then, following recommendations, apply a layer of clear hard wax oil. That would also work completely without sanding? As mentioned, no major damage is visible... but if the surface before painting is uneven (partly grayish, partly more like beech) – can I then even out that with the decorative wax, or will the differences still show through?

The difference is between "transparent" and "opaque". The transparent one is similar to a glaze, so you still see the old condition through it and the grain, thus also any existing grayish discoloration appears darker. That can look nice, depending on the condition of the floor before or after sanding, if you do that. If you decide for the "opaque" version, you might even be able to skip sanding in principle, if the current floor absorbs it (a small test will show you); otherwise, sand lightly and apply e.g. with a roller according to the instructions. In the past, Osmo had fewer but different "opaque" colors, and you can also mix them if a tone is not completely to your liking. As I currently read, there are for example "gravel", "silk", or "snow" as intensive opaque colors. Online there might still be other stocks, I once mixed mud gray with chocolate, and that created a kind of cappuccino....... I applied nothing more on top of this decorative wax, so that was the final coat, and that was never a problem in the living area (dining/living room). Your mentioned decorative wax oak is probably "transparent", so more like a glaze and more like a tint, so you still see the grain and discolorations through it, depending on the shade.

I can only agree with pagoni2020. Sand along the grain. In my opinion, use a hand belt sander. Start with a 120 grit and then do a test with the sealant; if there are still sanding marks visible, sand again with finer grit (250).

Ok, that sounds doable. Is a hand sander with a 180mm pad also okay, or is that rather bad? I happen to have such a device. How about the test? Won't it be uneven if I do just one corner first? Assuming it looks good... Then just paint the rest or how do I avoid an unsightly transition?

You are not removing any wood. If you want to then paint transparently over it, the surface should look relatively even in color. The "glaze" basically only changes the existing shade but remains mostly transparent.

Is a hand sander with a 180mm pad also okay?

The most feeling is in "your" hand but you can proceed cautiously. Of course, it makes more sense not to sand in circles or similar measures, but everything should be manageable for you. As I said, depending on the room, I would try it that way and then you'll see. Depending on the room, I would anyway paint it with an opaque coat right away, but that's a matter of taste. If you have pads with 180mm now, it's worth a try since the floor already has gray spots as you write. These color differences will still be visible after glazing anyway, but wood is never completely uniform.
In the end, it's no risk no fun, but in the worst case, you have to sand, which you intended anyway. Therefore, a try is no risk.

Then simply paint the rest or how do I avoid an unsightly transition?

You can see relatively quickly how it turns out, and since a glaze is not like a lacquered surface and you can see the grain/discoloration through it, there should be no visible transition. Try it on a spot where furniture will be placed or similar.
If you like it, you can still apply a second coat of glaze, then anything like that, if it occurred, will probably disappear.