I dated a guy like this for a very short amount of time. He constantly mentioned my background. Near enough every day, actually.
I am not 'posh' in the slighest- my parents were together, I went to uni and went to an ok-ish school (not private or anything). My family were comfortable but never went on holidays or anything, no big fancy cars, 3 bed house etc. We were not rich in the slightest. He, on the otherhand, was brought up in a typically 'rougher' part of the country, dropped out of school, has minimal qualifications and he had a very hectic family life.
None of his background had any affect on my opinion of him in the slighest. Why would I care? Why is it important? It just isn't. It's all about the person. However, for some reason, he really couldn't let my very normal background go.
Every bloody day. Posh comments because I went to uni (to study a fairly non competitive Arts course). Posh comments because I said something a certain way (because I was brought up miles away from where he was). Comments about me being up tight and a goody-two-shoes because I had my car taxed on time (???). He also mentioned in the past he dated a girl from the same town as him but she was from the 'posh' bit with a lawyer dad and a big house, but kept dragging out the point out of how 'well off' she was.
The whole thing was utterly bizarre. It was like an obsession. Needless to say, it didn't last long. He actually broke up with me (I was having doubts but wanted to see if things fixed themselves), but I think he knew deep down he couldn't pretend to be a better version of who he really was. We had had an argument about drugs a few weeks before (I'm very anti drugs. He made out he was but was seemingly changing his mind), and I think he realised that he wanted to go back to his single boy life with the lads from the same few streets as him. He never introduced me to his 'friends' (but he did spend enough time bitching about them and making me think they were all druggy scumbags in his 'I hate drugs' phase).
Honestly, I think he was a bit resentful that my upbringing was a bit easier than his. I think he saw stability in me and wanted to make himself 'better', but then realised a few months in that stability is boring and difficult to keep up when your heart isn't in it. I also am fairly convinced he cheated at the end as he went completely cold for a week before a very immediate dumping. He was completely uncontactable on a night where he had offered to pick me up after I had been out with friends until 4pm the next day. Pretty sure that's when it happened.
Anyway, I actually bumped into him a few years after the breakup. He was very chatty and 'nice', but he overshared far too much and it was a bit embarrassing hearing it, really. It was if his life never actually moved on past when I met him. Still owing money to people. Still having all these 'aspirations' but doing fuck all about them. Still trying to be a boy racer in his shitty car with an exhaust the size of a tree hanging off the back. I felt really awkward because I was pregnant (again, I'd just had my first baby with my partner and we'd bought our first home together a year or so previously). He knew about my first baby and was very shocked I was expecting another. I ended up saying little to nothing to him other than I was excited for baby and my other toddler was doing great. It just felt too awkward.
OP, I've no idea if your DP is anything like my ex. I hope he isn't. But even if he isn't, he does need to stop with the comments. He is putting you down and you should never feel ashamed of your background, no matter if it's positive or negative, rich or poor. He needs to accept you, exactly as you are. If he can't, then that's his loss. It's also not up to you to change him, though. He needs to stop the jibes on his own, without being reminded.
I hope you can figure it out and he gets a good grasp on what is really, truly, important.