This is an interesting and difficult question. The first thing to say is that the use of 'abstract' as a verb is uncommon in English. The past form, 'abstracted' is applied as a description of someone whose mind is on other things, or removed from the world around them. I normally expect that Danes know English at least as well as I do. The Danish word is closer to its Latin root 'abstraho' than the English, which reflects the supine form of that verb, which means 'I draw/drag away' or 'remove'.
The nearest I can get to your fascinating Danish usage is the noun 'abstract', which is an academic term for the summary of an academic paper or thesis or dissertation, which summarises the essence of the paper, without any of the detailed arguments beyond their general character. So it does not apply to the sort of thing for which you are looking either. It is something 'withdrawn' or 'taken away' from it. Of course the adjective 'abstract' gets us a bit closer. Abstract things are removed from everyday tangible reality. We talk, similarly, of abstract art, and we can certainly talk of discussing things or plans 'in the abstract', meaning in a theoretical way.
Looking at Roget's wonderful Thesaurus, I do not get much further. The verb can be used for to 'take away' 'shorten' or 'be concise'.
The full Oxford English Dictionary gets me a little bit closer to what you are looking for.
To withdraw, deduct, remove or take away something. So Boyle in 1685 gives us: "having <dephlagoned?> spirit of salt and gently abstracted the whole spirit, there remaineth in the retort a ???? substance." (my apologies that my failing eyesight and magnifying glass cannot get me to see every word in this miniaturised OED!).
In 1690, John Locke in his *On Human Understanding *is saying:
Truth and Certainty of Moral Discourses abstracts from the Lives of Men."
This is getting a bit nearer the mark, but is still not quite there and takes us back nearly 350 years!.
Better is the definition:
To separate in mental conception; to consider apart from the physical manifestation.
This surely is on the money for you. Citations go back the the sixteenth century and the latest is from the late 19th century, and they apply to logic and reasoning.
To abstract is to separate the qualities common to all individuals from the peculiarities of each individual. (Jevons Elements of Logic 1870)
So I am afraid I have failed and I cannot suggest an alternative word for the Danish. Could it be, I wonder, that this is one of those beautiful places where one language (here Danish) has a word that has a useful meaning for which another language (here English) has no equivalent?