I'm not so certain now having seen the pictures! At a guess, the floor has probably been re-varnished, and then waxed as well on top over the years? Again from the pictures, this looks like it has not lifted all the varnish off. Perhaps it has lifted some wax off, and a bit of varnish, or just wax?

I would forget my idea of trying behind the skirting as the varnish there will be unwaxed.

Is it is just the wax? Try some "Pial Laminat-Pfledge", Migros own brand, 6chf for a litre. Only try in on a 1" square area first. This stuff leaves a glossy residue that stops varnish sticking. If this is all that is needed, then I would start to feather the edges of the damaged area with a scotch brite or "mio brille" green pad, lightly, using washing up liquid and water as lubricant. The idea being to soften the contrast of the sharp edge left by the tape, make the damage a less regular shape. If you are lucky, cleaning will leave it ok, and this laminat cleaner will touch up the wax. You could then use that once a month over the whole floor to blend in.

Not just the wax?, then some 800 grit wet and dry paper, with water to lightly feather the sharp varnish edges, and some acrylic clear varnish for starters. If that leaves the new areas tool light it might be clear polyurethene varnish, although I doubt this.

It looks like the floor needs renovating anyway (at least by Swiss standards), so hopefully you will get away just with the cleaner.

This is the kind of conundrum that definitely has me going to visit my local chemist (drogerie). They always seem to have the right solution to these kinds of problems.

Just a thought but would your insurance cover it? I did similar damage to our wooden floor and 'accidental damage to a third party's property' was covered. In the end the landlord didn't notice it but the insurance would have covered the proportion of the repair bill we would have had to pay.

Unfortunately (or maybe fortunately) it is the floor of my partner's house so the insurance side of things would be a bit complicated.

The floor will need to be re-done at some point but since you can only do it a couple of times (take off thickness each time), I am hoping to find a solution that will make it a bit less obvious in the meantime without creating a further mess. I will do a bit more research this weekend.

My guess is that I took off some wax but also some of the varnish.

Thanks to all of you for your suggestions.

__________________
Do or do not, there is no try(ing). Yoda

Although you'll probably be reluctant to do the same again after what has happened, the next time you feel the urge to put tape on the floor, use low-tack masking tape available from art material suppliers.
It won't lift the surface.

And believe it or not, I thought that was what I was doing with the blue tape. Now I know.