The "[noun] de [person]" way of indicating possession is just as grammatically valid. For example, a more specific way to say the sentence with "your" might be: Él necesita abrir el correo electrónico de usted(es). To be fair, the English words "it" and "they" can be just as vague and can be replaced with more specific phrases in those less certain scenarios.
For court transcripts specifically, the stenographer can put in descriptive phrases like "Indicates the defendant." That's what's happening when someone in court says "let the record show that..."
I´m surprised no one has asked why correo is pronounced with the e stressed even though it´s not not written that way. I have verified that this is the correct spelling and pronunciation (https://forvo.com/word/correo_electr%C3%B3nico/), but I don´t know why. It seems like there should be a list of such exceptions somewhere.