When you do your PhD, you‘re not completely on your own, you have a supervisor. And then maybe a PhD committee and a second and a third reader and so on. However, during the process of developing your research question, diving into literature and possible approaches, making discoveries, and finding a place for your future academic persona, you primarily interact with your PhD supervisor. He or she should act both as a coach and as a mentor to support you on your academic adventures.
Interestingly, in German speaking academia, your supervisor is still rather called „Doktorvater“ (doctoral father) or „Doktormutter“ (doctoral mother). Which implies a more family-like relationship. And which also is in line with the traditional notion of not studying somewhere at a certain university, but to study with someone, i.e., be the (graduate) student of a specific professor. And thus become a member of a specific „school.“ In the old days, the members of an academic family stood together, supported one another, helped with getting promoted, etc. Which is what you would actually also expect from a mentor. So the role somehow fits.
By the way, how are the PhD students called, I‘m not aware of a label as „Doktorkind“ (doctoral child). You are the „Doktorand“ (male) or „Doktorandin“ (female) (doctoral student) of someone. However, this is derived from the present participle of the verb meaning „to do a PhD.“ Which means, after the defense of the thesis, the label doesn‘t fit any longer. You might be a „former PhD student“ of someone, but this person doesn‘t turn into your „former PhD supervisor“ or your „former Doktormutter.“ He or she keeps the label and thus probably also the role, even after dozens of years.
Surprisingly with this family notion, at least on the side of PhD students expectation grows that these ties will last for longer—you cannot get rid of fatherly or motherly duties—and that mentoring or coaching support also will last for longer. So they tend to get disappointed when mentoring-like support stops, no information on (future) projects or even jobs are passed on, no reference letters are written any longer, and so on. Of course, one could argue that during your PhD you should also find your own way, stand on your own feet, and leave your supervision family to start your own. And of course family relationships aren‘t always positive, there is abusive behavior which is hard to report and will stay within the family. And as long as everything looks great from the outside, nobody will believe that the inside isn‘t as bright as current incidents at the ETH show.
Another aspect seems to be gender, actually. And maybe more on the side of the supervisor. Female supervisors (so the „Doktormütter“) seem to be more protective and more supportive, at least they report such actions on social media and they even use selfdescriptions as „mama bear advisor“ and the like. And from what I see (which is obviously a very small snapshot), more female supervisors state how proud they are when their current or former PhD students report success stories (an award, a talk at a prestigious venue, a good job, another grant, etc.). Male supervisors also show success of their PhD students, but with much less emotion, they rather mention this as a success story of their lab/institute/project. Which fits stereotypes of motherly and fatherly support within families, so the German terms actually are apropriate, don‘t you think?