My mom always encourages me to get my jewelry in sets, when possible. She is a HUGE fan of this and over time I have seen how beautiful and polished it looks.
Most of my designer pieces I try to get in sets. Sometimes I wear them together, other times I mix and match.
I think your Bzerio necklace and earrings would look beautiful together and not "old" or "old fashioned" at all.
My mom is the same way too!!! Even though the concept of matching sets is from an older generation, it doesn't come off as old fashioned. The best part of having a matching necklace to your matching earrings is that you have the opportunity to wear the two together when you want and the ability to wear one, the other, or mix it up!