I’m fairly sure you choose it yourself. I believe you fill in a form at the municipal office. I think it has to match what you have on your passport though. The rules may not be that strict though, because my friend studying in Japan was telling me that he technically could have written his name in kanji, just that he was worried it would cause problems with proving that documents belonged to him. (We’re both ethnically Chinese, so we have Chinese names along with English ones. Both names are on official record in our country of origin, but we generally use our English names/name transcription while overseas. Our passports are also only in English.)
I have no idea how aliases work since all of my names are fairly short. I’ve heard the same thing as you (that aliases can be officially recognised), but I don’t know exactly how it works.
EDIT: Here’s a document from the Ministry of Internal Affairs covering how aliases (通称名) work. You might want to download it and rotate the second page though, because it’s a headache to read in its original orientation: https://www.soumu.go.jp/main_sosiki/jichi_gyousei/c-gyousei/zairyu_1/pdf/080805_1_si1.pdf
Short version: it seems you can register just one alias at the municipal office. I’m not sure if you need to provide proof that it’s being used though. I don’t really feel like analysing the document in full right now. Sorry. Plus, I think someone with experience would be able to give a clearer answer than that document, because I’m not sure it covers the procedure.