Why would this have a generic answer? It depends on the people and the specific case.
“Can someone pursue a relationship with someone who is wearing another person’s friendship ring?”
– What?
People do not have universal codes about things. Especially not friendship rings. Such meanings are fundamentally arbitrary and individual.
It seems to me that one of the most common causes of relationship upset, is people thinking that what they think is going on in a relationship, is the truth, without asking the other person. And/or, that there is only one set of meanings that apply to all people in the same circumstances.
One person’s Platonic friendship ring is another person’s sure sign of romantic interest. People need to talk each other, listen, and be open to other people having all sorts of different perspectives and meanings, to have much chance of agreeing on what’s going on in a relationship.