Does concavity of the utility function has any bite? A utility function in general has only ordinal meaning, any monotone transformation preserves the order isomorphism of the underlying preference ordering. However, there are several econometrics papers, that estimate demand with concavity shape restrictions on the utility. 
Is this justified? 
My first thought is that yes, that the key idea is that there exists a concave utility representation, and this generates the dataset, however it worries me the identification issue since there is infinitely many of these utilities not all of them concave. 
In short, does it make sense to require concavity restrictions on the utility function in econometric work?
This post shows clearly why in the world of "standard" ordinal utility, concavity of a utility function cannot obtain an economically meaningful interpretation, although it may be useful as a mathematical property.

But "standard" ordinal utility is not compatible with Econometrics, because Econometrics deal inherently with situations where there exists uncertainty, and in a framework with uncertainty we move from "fully ordinal utility" to Expected Utility theory, where properties like concavity have economically meaningful content -they express attitude towards risk (as well as "preference intensity", as this laborious post shows).  

The concavity assumption/restriction is made because there is universal consensus that the vast majority of people exhibit risk aversion in their economic behavior.