One can imagine growing up as a child and hearing Japanese with its variety of first-person pronouns, which also are not used as much as we use them in English, and choosing to use your name as it is consistently associated with you.
That said, I've had only a smattering of Japanese study, so definitely take all of that with a grain of salt.
When you're regularly calling everybody else by their name, calling yourself by your name makes sense.
However, do keep in mind this is anecdotal so I don't know how much it applies to more broad settings.
Actually, calling the other one by their name + san/chan/kun is the most common way. Using "anata" (= you) can even be considered rude under certain circumstances. My wife never calls me "anata". Also, pronouns are often omitted if it is clear who is being meant (a bit like in Italian).
Using or not using the pronoun has the same emphasis as saying to someone:
'You're drunk.' versus pointedly saying 'You, are drunk'.
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Soy Latte, por favor! (In Spanish this translates literally as 'I'm a latte. please.')
I'm not a linguist, but it seems like there's no grammatical difference between first/second/third person in Japanese - you can consider all the sentences to be in third person.
If not their first name, then their title "younger brother/sister" (adik), "older brother sister" (kakak). Interestingly, when the second child is born, the "adik" becomes the "kakak", which you would think would be confusing but the switch happens pretty naturally.
Also, in case of adult women, if they speak in third person, to me it sounds あざとい(azatoi). I am sorry, I don't know how to phrase it in English without ruining the nuance. It's a mix of calculating, cunning, and a bit flirtatious? Either way, you also need to consider the tone of their voice, body language, and the contents itself. (Maybe if they want you to carry the shopping bag because it's a bit heavy for them, they might use this way of speaking.)
That being said, I do not have any real-life examples to back this up so take it with a grain of salt.
The closest word I can think for what you describe is insincere. See also: almost every influencer on TikTok.
Going between languages is hard. :)
To people that actually do this, do not erase yourself; you are a person, you have needs and an identity of your own. You can express your needs, not a mythical 3rd person parental figure.
Honestly, I think its a great way to get some perspective (but I am on the spectrum). I also find it plain fun, and -sometimes- cute. Doesn't mean I always do it in family circle. Its just that, in a work environment, there's an etiquette, and this one isn't part of it. If you're too different (e.g. weird in this context), they don't want you, and one way or another you'll get cast out. Goes for weird people too.