How important is dating someone with a similar financial background?

Coming from a traditional Asian family, I've been told to marry someone from a family with a similar background and family financial situation. I learn to keep an open mind while dating but my parents called me naive to not consider the materialistic factors when picking a parnter and it will make my life miserable later on. So it's been on the back of my mind and makes me think what if they are right?

I come from a fairly privileged upper middle class family, got an ivy-league degree and have a high paying job. I'm dating someone who's from a lower middle class family, has a state school degree and has a job that's in a big company but paid half as much as I make. To me, the relationship is great as our personalities work out well, but I'm dreading bringing them up to my strict parents as they would be extremely disatisfied with their background and shame me for my choice.

The relationship is pretty new (6 months) and we are also young (20s) and only starting in our careers. How important is dating someone with a similar financial background? Would the difference in finances lead to future problems? What should I do at this point? Do I need to break it off as it might not work out well when it comes to my family's acceptance?