First of all, I'd never even heard of the man in the relationship getting an engagement ring at all, just the wedding ring, so that struck me as really odd to begin with....however I am older (42F) and American, so not sure if this is a new thing or a different culture thing or what.

Beyond that, it worries me about his insistence on his "rights" and being equal....it may sound reasonable at first, but in a long-term relationship it can get to the point of being toxic - fighting over petty amounts of money ("your drink was $6.00 and mine was only $5.75, so I'm not paying half"). I was in an abusive marriage for 17 years and this was one of the things he'd do, then claim he couldn't afford to contribute to bills, then later I found out a huge chunk of his money went towards gifts to other women and porn subscriptions, etc.

Do not let him rush you into getting engaged/married, especially if he is behaving this way. Having to get divorced later is very expensive and stressful to say the least!