No it's not. You can change your class you can't change your caste - your caste is literally determined exclusively by birth. It's a form of race and has nothing to do with social standing.
It’s even more interesting how norms and customs predate even caste. For instance, missionaries came to India and offered a better status within Christianity to Hindus from lower caste. This led to some caste based discrimination against Indian Christians. Now, Pakistan as a country doesn’t really have caste system being a Muslim country. But there are restaurants in Pakistan where Christians are still not allowed to eat in the same plate regular customers eat!
Source: A White Trail: A Journey Into the Heart of Pakistan's Religious Minorities Book by Haroon Khalid
Isn't social class specifically evolved to not simply be traversed by attaining more money, and that's the whole distinction between "new money" and "old money" and how they say "money can't buy class"?
That being said, I think perhaps the caste system is sort of neither about race nor class but some unique blend of those things and more. It seems to have its own characteristics and I'm not sure we can dumb it down.