It does sound really weird but then again, lots of engagement rings look pretty similar so I'd be inclined to take it as a compliment - how often are you ever going to be comparing rings? Unless the stones are something genuinely unusual, there are more and more engagement rings with less common gems like alexandrite or bentonite or tanzanite etc nowadays so I think you just have to chalk it up to the fact that people are branching out from the usual sapphire, emerald etc and the only way people learn these things exist is from seeing other people's rings with them! So, a bit annoying secretly, but not that bad.
Also, the engraving thing would be very weird if she has copied your nicknames for each other or something, but if it's a common phrase or music/pop culture reference (like, I don't know, "you're my person" or "to the stars" or "I love you 3000" or something) then I think YAB slightly U.
At the end of the day, your ring is special because it was given to you, as a symbol of your relationship, by your partner. What it looks like is not what makes it special, nice as it might be - people have some fuck ugly rings, and millions of people have almost identical solitaires, but they're still special to them. Jewellers, even fancy ones, sell tons of the same style of ring every year.
Yours is still just as special and important because of what it symbolises, not whether or not someone has the same one.