Is there a name for this phenomenon?
There are several in fact, but there doesn't seem to be a single unified term, which is quite a problem because it makes looking it up a real pain in the neck.
Amazingly of languages that have this feature I have yet to see one have a specific native name for it. I myself as a Turkish speaker describe this to people around me as "when your mother calls you 'my mother'", but that's about it. The very demand to name it seems to have arisen from those who study it, who often don't seem to have such a common occurrence of the concept in their own languages.
Anthropologists (especially Americans) often call it self-reciprocal kinship, although that term can be extended to given names and honorifics depending on the author's perspective.
[...] What may be termed conceptual reciprocity is an exact accord in range
of inverted meaning of the terms for two relationships. [...]
The generic terms talakyi and ulani are each self-reciprocal; but they are
hardly terms of relationship in the strict sense.
(Kroeber, A. L. (1919). Zuñi Kin and Clan. Anthropological papers of the American Museum of Natural History, Volume XVIII, pp. 79-80)
Psychologists don't seem to have a specific name for the term, but I have encountered (at least the quotations of) a few who describe this action as "a method to teach children how to address their older relatives by anticipating them to repeat".
And finally linguists seem to use a number of different words depending on the focus of their study.
Affectionate kin vocatives are social honorifics that are employed to en- hance intimacy among relatives and acquaintances. Jordanian Arabic features several of these, e.g., yammah 'mother', yābah 'father' [...] these affectionate kin terms are used in two distinct ways. First, parents, grandparents, uncles, and aunts may use their own designated absolute honorifics in addressing their respective descendants, thus reversing the denotational signification for the purpose of showing affection. For instance, Jordanian mothers usually address their sons and daughters as yammah 'mother' instead of yabni 'my son' or yabinti 'my daughter' for various purposes in interaction (e.g., summons, requests, questions, etc.), as in the utterances in (9) below.
(9a) 'a tf-ni may yammah
(mother-child) give-me water mother
'Give me water, son/daughter!'
(Farghal, M. & Shakir A. (1994), Kin Terms and Titles of Address as Relational Social Honorifics in Jordanian Arabic. Anthropological Linguistics, Vol. 36, No. 2, pp. 246-247)
Reverse addressing is the possibility of addressing a recipient with the addressor‟s kinship term and try has been made to investigate almost all its aspects based on harvested and self-reported data. Since this feature is not a universal feature of all the natural languages, representative examples are used with the English equivalents to provide all the readers with an in-depth understanding of this phenomenon.
(Tavakol M., Allami H. (2013). Reverse Addressing in Modern Persian. International Journal of Society, Culture & Language, Vol. 2, Issue 1, p. 37)
How common is it?
That's a bit tricky. Depending on the language, country or region this phenomenon can have several connotations. While a primary way to address relatives regardless of age in some cultures, in others it's exclusively used for children and in yet others as a form of belittling. As a result, this itself is a sociological question in its own right.
Next to consider are the many variants which can even differ for speakers of the same dialect of a language:
- As briefly alluded to above, the "relatives" this can be applied to don't only depend on which relatives have a word designated for them.
(ex. A Bulgarian uncle might not call their niece an вуйчо (vuycho) ~uncle, but their grandfather might call his granddaughter дядо (dyado) ~grandfather )
- Sometimes the gender of the addressee can be the reference rather than the addressor (ex. a Mexican mother might call her son papá instead of mamá)
- "Vertical" or "Horizontal" generational extensions are possible.
(ex. A Vietnamese mother calling her daughter cô is calling her an aunt, rather than mother)
Is it confined to a geographical region?
Now you're talking! This is actually a pretty interesting question that I don't think has been covered in-depth before, so I've spent the past few months surveying people across the internet to make this map:
In a nutshell: It's common throughout Latin America, A good chunk of Africa, The Balkans, The Middle East; more region specific for Italy and South Asia; and appearing in pockets throughout Western Europe, Central Asia, The US and Oceania.
See the above link (or click here if it's invisible for some reason) for a more in-depth analysis.
In short, how can I find more information about it?
if any of the above citations were an indicator, anthropologists might be of more help to you
I myself personally recommend The Genius of Kinship: The Phenomenon of Kinship and the Global Diversity of Kinship Terminologies by German Valentinovich Dziebel. It's a pretty good source covering many disciplines but best of all it proved to be a very good starting point for my own research on the subject.
Any and all of the above citations, and references found therein. You might need to be a bit archive-savvy to find those American Anthropology Journal articles but the later ones should be a bit more accessible.
If you are particularly interested in my map, you could try making an acquaintance from a particular country and asking them first hand about it, amazingly this is a lot more effective in distinguishing nuances between how different cultures implement their kinship reversals.
I hope that somewhat clarifies things for you. Feel free to comment for any further clarification.