I don't just mean the alphabet song, which originates from the US BTW. I mean the linguistic phenomenon which I cannot find the name for because I cannot put together a search query that isn’t too generic. The most obvious occurence is know one’s stuff or know one’s [general subject of study] but it occurs with other verbs too. The 8-Bit Guy recently said “here are your cursor keys” when describing the odd layout of an ‘80s keyboard that nobody is using nowadays so the pronoun “your” seems inappropriate. I complained to linguists and they didn't take me too seriously (presumably they just let the language evolve without considering if it makes sense, which I guess will eventually result in “your” being officially recognized as an alternative spelling of “you’re” – a mistake almost only Americans make).
And yes, I guess Brits say “know your ...” too but I associate that with Americans. And note that phrases like “know your place” or “know your neighbor” are exempt for obvious reasons.
Take "your keys" vs "the keys". In both cases you have a determiner and a noun. Now because European languages are not topic prominent they mostly attach some kind of a determiner to nouns. Now most language in Romance and Germanic family over several century pick "the noun of noun" while English went with more "adj noun". This results in English using articles less then other languages.
Neither of these is more logical then the other. IT just each language need some way to specify the scale of nouns. I am talking about all keys everywhere, no I am talking arrow keys. In a lot of langues I wouldn't even use the word "the" when talking about them because many languages just don't have a word for the.
Maybe in Czech (guessing from your username) they don't use possessives nearly as much? To me it sounds absolutely fine. In your example it could be seen as him showing you the keyboard (like your boss showing you your cubicle or whatnot). Funnily enough, in my native Spanish we also use possessives a ton in pretty much the exact same way, so in English it seems normal.
Sounds like a language issue you're having with English in general since as much as I like to mock the Yanks bastardised English, what you're describing above sounds fine to me as an Englishman.