Is it possible to automatically obtain cited-by lists? In google scholar, it is easily possible to obtain a list of articles that cite a specific article. Web of science and scopus also offer this feature, if you manually enter the article identifiers (e.g. doi or database identifier).

However, to automatically (api-wise) obtain a list of articles that cite a certain reference (cited-by) seems to be very hard. I'm able to query metadata via the respective APIs of web of science and scopus, but the cited-by features seem to be fenced. For crossref, my credentials are not sufficient as well.

As far as i can see, only pubmed offers this option, but my field is not related to (bio)medical applications.

The point is, organisations giving you access to do this by hand apparently do not like you to do it automatically. This is information that is not behind some paywall so I do not understand. 

Q: Has anyone bumped into this as well, an explanation for it, and suggestions to get access to a list of citing articles in a programmatic way?
Try OpenCitations.
Using OpenCitations' COCI (guideline for their REST API here), you should see all references to {DOI} via https://opencitations.net/index/coci/api/v1/citations/{DOI}.
Example: There are 243 works citing the paper 10.1177/03058298020310031201 (according to CrossRef: is-referenced-by: 243). Use this DOI in COCI to see all these 243 references DOI-by-DOI.