JStewart (Bohemia Interactive - administrator) told me "On the public hive there are two types of servers: 1st/3rd person servers and 1st person servers. Player characters are individual for those two types of servers and are thus not shared across those two server types".
This was news to me as I thought it was normal to have just one public character that gets the server hopping delay you would expect when moving between first and 1/3pp servers. So I asked the question on Reddit and the replies were divided into two groups and each group thought their experience was normal.
Those who have the usual two public characters seemed to think it should be reported, I guess it is not what first person players and server renters want happening.