All that is wrong is that the Project file does not have the correct binding to TFS. There is a section missing in the .csproj file that should look something like this:
<SccProjectName>SAK</SccProjectName>
<SccLocalPath>SAK</SccLocalPath>
<SccAuxPath>SAK</SccAuxPath>
<SccProvider>SAK</SccProvider>
Where SAK (probably) stands for Solution Already Knows.
As far as I know it only affects the display of the Source Control status in the Solution Explorer as noted you can already perform other source control tasks on the project. It's not much but worth fixing as it's a good visual check that everything is being controlled.
To fix it highlight the project in the Solution Explorer, then go to File > Source Control > Advanced... > Change Source Control... Select the project in question in that list (it'll probably look fine and valid), click Unbind, ok the warning then click Bind (the same toolbar button relabelled) Ok that window, and checkout all the files it suggests. This will make the necessary changes and the padlocks should appear. Review and check the changes to keep them.