I think you're mistaken there.

Wine is a vanilla Linux executable that runs as the user who launched it. The Windows program it runs thus also runs under that user. That's possible because Wine doesn't do anything system-wide (like intercepting calls or anything), it already gave the process its own version of i.e. LoadLibrary() (the Windows API function to load a DLL) and can happily remap any loaded DLL to Wine's reimplementation of said DLL as needed.

Here are, for example, the processes created when I run Paint Shop Pro on my system (the leftmost column indicates the user each process is running as):

Also, some advice from WineHQ: