Player search and linkshell/Free Company lists also do not identify the class icons either if you hover the cursor over them, so for a new player just starting it, it could indeed be confusing.

And although most of the class icons are fairly obvious what they are, Job icons however can be far more obscure or abstract, especially if you're unfamillar with traditional FF Jobs (like, how many non-FF fans would know what the SMN icon is meant to be? It's meant to be it's hat, a horn which SMNs often have as headwear, or, in the case of one FF game, as an actual physical trait), so I agree with the OP that this would be a helpful addition to the game for new players.