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.
I think that the way marginal utility is frequently used is that there is an assumed unit of measurement. For example if in the utility function $$ U(x,y) = v(x) + y $$ $y$ denotes income spent on other goods then the unit of measurement becomes money, as one unit will always increase the utility by one. In this case $MU_x(x,y)$ is actually $MRS(x,y)$ because $MU_y(x,y) = 1$, so the two are equal. The concavity of $v(x)$ will mean that the preferences w.r.t. $x$ and $y$ are convex. This remains unchanged if you perform monotone transformations.
I am afraid I cannot give advice on econometric applications.