@definitionofinsanity
You didn't mention anything about them having a previous relationship...
Umm,
ultraleaf said "those two have met before(the parents)", and i responded "Well, obviously". I think that counts.
Think about it. Doesn't the tone and what he says sound more like people that know each other pretty well instead of two fathers, one of them being so distant he's barely even in his own (step)daughter's lives, let alone mingling with the in-laws? I mean what Fuutarou's dad says is basically the generic Japanese manga/anime motivation lines that a friend says to another, really.
Yeah, they are pretty friendly in the flash-forward. But, considering it happens in the future and the fact that Fuutarou's dad was his usual laidback self and Papa Nakano was his usual distant self, it doesn't automatically follow that they knew each other long before the events of the manga. That said, it's not that i think that wasn't the case nor would i be surprised if they actually go way back together.
Sensei is a general thing that ties in with Japanese honorifics...
And that's why i said it was "vague". Though, it kinda slipped my mind that he indeed worked in a hospital, so that makes sense, and doesn't really explain much about their relationship.
Actually, i think i made a wrong assumption - i thought Fuutarou's family wasn't aware of him being fired from tutoring job and that Papa Nakano said nothing to Fuutarou's dad about that. Now that i've put more thought into it, that would have been strange. As an employer he most likely contacted Fuutarou's guarantor (am i using the correct word?). Based on how their talk played out that actually might have lead to their current act.