Is My Dad Your Dad or My Dad?

I had the following conversation today:

'I think I know your dad.'
'How do you know him?'
'My dad knows him.'
'And who's your dad?'
'He worked with your dad at the college.'
'What was your dad's name?'
'The same as mine.'
'Sorry, what was your name again?'

My brain was hurting even before I lost the will to live.