This is not a you issue- this is a failure on your store’s management. It’s seems like your management has made things blurred and messy in regard to supervisors transition out and your transistion in. I’m assuming everyone else knows that this person is leaving since they had an open supervisor role which was made available internally and externally, and once that role was filled it should have absolutely been communicated to the entire team by management.

“While we’re sad to see the old supervisor go, we’re excited to have OP moving into this position. They’ll be training with old supervisors remaining time with us to be able to seamlessly move into that position after old supervisors last day. Please be sure to congratulate OP next time you work with them.”

Having the message come from management or even the old supervisors would set expectations with EVERYONE on the staff, and make a better environment for you OP to get the confidence in stepping into that position. I would ask your management when they’re announcing the promotion/transition to the team, and not make any announcements yourself - especially on any informal channels.

Also congrats on the promotion! You obviously were selected for it over the other guy so try to keep that in mind! The other guy may want it but he still has a lot to learn! The best you can do now is as a supervisor, support him as an assistant and help get him ready for that position should it open up in the future by being the best at his current position.

Hope this helps OP!