I had a PhD interview (Computer Science) 3 weeks ago with a German institute. The interview went well and they said they will make their final decision in a week (the latest) and let the candidates know. However, it's been 3 weeks and no response. I emailed them a week ago asking if they have decided yet. Still no response. Before the interview, they were extremely responsive. E.g. I emailed and asked many questions about how I should prepare for the presentation etc. They would typically respond in like 5 mins. But after the interview, they seem to be ghosting me. Does this mean I should forget about this position?
It could be that unforeseen circumstances have prevented a timely decision. However, most likely you are not their first choice but they don't send out rejections before their first choice has signed the contract. That's pretty standard.
Something similar happened to me in the past. Three months later they called me and asked if I would still be interested. Their first choice had decided against the position. If they had send out rejection letters they would have had to restart the whole process again (with a new job posting).
You can't actually conclude anything. There might be many reasons for a delay and a lack of communication. One reason might be a time of negotiation with another candidate and holding on to the others in case they can't come to agreement.
If you were the first choice then chances are you would have heard by now, but the delays might be just administrative and seeking internal agreement.
What you can do, however, is keep any other options you have active and open.