What is bearnaise sauce thickened by?
The thickening in Bearnaise, as in mayonnaise, is not so much in the ingredients as in the technique. These sauces get their thickness by being emulsified.

An emulsion is formed by rapidly mixing, whisking or blending two ingredients that shouldn't mix (oil and liquid). The emulsifier (egg yolk and often mustard in the case of mayo) stabilizes the emulsion (tiny oil droplets suspended in liquid) in much the same way detergents break up grease.

This wiki article goes into far more detail.