What could be the reason that there is a smell coming from the shower drain like the sewer is backing up? :eek:

Backing up, or not fully draining in the first place. Kind of the same thing. Has it been snaked? How old/new is the install?
Is the shower/tub taking longer to drain? There may be congestion in the trap and that trap needs cleaned. There are several products on the market to facilitate the flow issue, but I'm certain that some of the seasoned pro's here have some home remedies that could help you.

Is the p-trap from shower siphoning (water drops below trap seal).. Is there any bubbling. Occurring? Need to look for this type stuff... Then try to flush toilets upstairs and see if shower bubbles... or run washing machine if that is closest.. you know... try to flood the system a bit and all the while have someone watch drain and see/listen to what happens at shower.

Shower drain may be clogged... main drain could be clogged... tell me... 1st floor, second floor.. basement,. Get back to us after you try these things.

The most frequent cause of odor from the shower drain is fermenting hair, skin cells and body oils. Its you, you're down in that drain rotting away. Mix 1/2 cup of baking soda with 1/2 cup table salt, dump the mix into the drain and then add a cup of vinegar, stand back. Tomorrow wash it down with a big bucket of very hot water, smell better?