To answer the title “What Does Fatherhood Mean…?” – as all words – their meanings are an amalgamation of perception/s.
The quintessential question then is, what does G-d teach?
Every day we are commanded in the Shemah – “You shall teach the love (and reverence) of G-d…”
In other words – and this is critical – a father, exactly like our heavenly one – is a positive role-model who teaches. (Unfortunately our modern society has confused democracy with anarchy – it has always been the privileged, duty and pleasure to pass positive guidance – and as the Rebbe teaches, doing so, not only does not take away your child’s freedom, it – considering the insane deleterious influences (perhaps never before, that children were ever so exposed, in countless mediums) it is literally saving, both their soul/life.)