Can the user of the computer access any information they want via the Kernel on the underlying system running Wine, or does normal Windows security still apply?
Well, Windows security applies to the extent that Wine emulates/implements it. So this depends on what exactly you mean by "Windows security".
What definitely will apply is the security of the underlying OS that you run wine on.
To the hosting OS, wine is just another application, and all restrictions WRT filesystem access, net access etc. apply. So a Windows EXE run in wine can do all the things a regular native program can. So if you want to restrict an app running in wine, the best thing is to run wine under a user account with limited privileges, or in a chroot (if the hosting OS has this).