It's a common pronunciation of this word:
'i-nyuv' is common to Munster Irish
'i-nyou' is common to Ulster Irish
'i-nyuh' is common to Connaught Irish
All three will often be heard on Irish radio.
I think 'schools' Irish (learnt Irish), at least in Dublin (or Leinster) uses Munster Irish in general but where pronunciation is easier (closer to its spelling) in other dialects (often Ulster Irish) those pronunciations are adopted, creating a hybridised fourth dialect, generally with English phonology, which hasn't had positive consequences for the spoken language.
I think the 'ubh/uv' sound you are hearing is due to the fact that 'inniu' has developed from old Irish, and it is effectively, two words, like 'to-day', a preposition and noun, so that sound expresses the obsolete dative case ending for the noun 'día' (which was used to also mean 'day' in the past, and not only, God, like now). Perhaps, it was dependent on the function of the word, whether 'noun' or 'adverb'.
Below are the various forms (identified as having been written) of 'inniu' in Old Irish: eDIL s.v. indiu or dil.ie/28446
Forms: indiu, indiu, indíu, ani, anniudh, aniugh, aniubh, aníbh, anú, andiumh
Is there anything wrong with, "The day's date today"? I could see it literally is, "date of the day today" with "lae" being the genitive, but that doesn't sound like native English. Their other answer, "Today's date," merges "day" and "today" so loses some of the original. However, "Today's date," does sound like native English to my ear.
But aren't 'today' and that 'date' definite? To me the Irish makes sense as in 'the date of the day today' when literally translated back to English. But English often has a preference for apostrophes to show ownership. Maybe the discussion here will help: https://forum.duolingo.com/comment/10814496/When-to-use-the-definite-article-in-the-genitive-case