OP if we met you in person maybe it would appear you’ve (not in a desperate way, just in a natural way) “switched classes”? 
I agree with pps that the degree/career path has an element of relevance, but it’s not necessarily automatic?
I know two men who were “first generation to go to uni.”
Both had parents who would solidly identify as working class (tradesmen). And definitely identified that way when younger.
They studied hard, traditional subjects at good unis (not Oxbridge and medicine/law but “solid” STEM subjects).
Went into military at officer level. So they had that experience of university socialising, then thrown into an environment where there’s an emphasis on “being refined”.
Hard-working blokes, eventually married to university graduates in professional jobs (maybe each brining home 50-70k in their 30’s).
So financially stable, well-educated, dress a certain way, not too “flash”.
I think due to the degree AND the “doing the traditional job path” they sort of moved in their thirties from working class to “middle class with a regional accent”? Their children will be middle class.
Alternatively, maybe they could have made a lot more money by graduating then putting their practical and numbers skills into buying and doing up property. But I don’t think they’d have necessarily been as middle class then?
(Class is a bag of made-up shite really but it does exist - I’m planning on moving abroad btw and I can’t wait
)