What does non-exclusive publishing rights mean?
I suspect you're seeing "non-exclusive distribution rights", which means the journal gets the right to distribute the paper, but it's not an exclusive one. Other venues, such as arXiv or ResearchGate, can also distribute it.

If you have an open access paper, presumably it uses the CCBYA license, which means any use, distribution and reproduction in any medium is permissible (i.e. you can publish it elsewhere) as long as the original authors are credited. Journal licenses that only allow the authors to self-archive are more restrictive and usually indicate that the paper isn't open access.

This doesn't mean you can publish your article in other journals. Most journals will have policies that all submitted content must be novel, and if you've already published the paper it's no longer novel.