Is the recipe for my shaving cream complete nonsense?
As @Mithoron commented, these are ingredients, not a chemical breakdown of the result.

And, as @Karl commented, traditionally you mix together some rendered fat or oil -- that's a mixture of fats and oils with the fatty acids and glycerin that make them up -- with potassium or sodium hydroxide to make the potassium or sodium salt (conjugate base) of the fatty acids, which is soap.

Here, you've got:


Solvents: 


Water
Limonene

Fats, glycerin, and fatty acids:


Stearic Acid
Myristic Acid
Coconut Acid
Glycerin
Theobroma Cacao (Cocoa) Seed Butter
Melaleuca Alternifolia (Tea Tree) Leaf Oil

Base:


Potassium Hydroxide
Sodium Hydroxide
Triethanolamine

Extra smells


Parfum (Fragrance)
Also Limonene 
Linalool
Also tea tree oil



The only reason it has so many is that they've mixed together either a few different soaps or a few different fats before saponification (soap-becoming), and each soap will have been (mostly) neutralised.

Edit: note that the cocoa butter and tea tree oil are almost certainly added after saponification, for fragrance, to alter the material properties of the product, and to be left on the skin.