Anything can be secret. This is a feature I see on any kind of site that has a userbase, forum or otherwise.
I would instead suggest that this be a per-user option. Some may wish to have their profile not available. Some may wish to have it open. For instance, a few roleplay sites I go to have this option, and even have the option of excluding my profile from google (no index, no follow, ect) while still technically making it viewable to the public. This is because I feel, as a single person, I would like to exclude the rest of the world from what I’m doing on a single site. In my case, I would rather share my stories with those who have accounts. For the ‘exclude from google searches’, that allows me to send a link to my profile and let others view when I wish. Of course, if someone really wanted to find it, they might be able to. These deterrents are for those casual, random eyes (i.e., the world).
Other uses include any sensitive information on the profile as dictated by the forum’s topic as @ZeroFlux and @Barbie explained above.
Here is a sample from the RP site I go to: