Identifying check-in sender with SMS?

I just activated my inReach Mini 2 and sent a test check-in to my brother via SMS.

His screenshot back to me shows just the check-in text "I'm checking in. Everything is okay." and a link inreachlink.com/...

It doesn't show my (sender) name at all, so it looks kind of vague or suspicious to folks who are jaded about scams/phishing etc.

Is the random sending phone number supposed to be stable and only used for messages from me? It's not clear to me whether he's supposed to add this number to his phone contacts to assign my name to it, or if it is some generic relay number that could be reused for other inReach senders too.