My opinion is not yet fully set on the so-called "ethnic" or, as they say in the US, "racial" statistics.
From an empiricist' perspective, the more data you have the better, whatever its quality. Indeed there are ways to "clean" data from mis-categorizing.
Hence in order to understand how discriminated against some groups are, it is important to record as much information as possible.
Yet, I fully understand the impact that having to categorize oneself can have. One of the big suprises, as a French person, I had in the US was how stereotyped people were. I heard continuously expressions like "this is so white", "this is so black" and so on. I think that having to define oneself as belonging to a given ethnic group may force people to identify with a group they do not feel anything towards.
But, at the same time, I feel that this is a "majority feeling", I am inclined to believing that it's members of majority groups that have the impression that naming subgroups may force them to take an existence that they wouldn't have otherwise. It is very clear that when one feels discriminated against, one does not need to be assigned to a group to know that this group exists.
Hence I suppose, there is more evidence supporting the use of such statistics. But I do not work in this field so I am probably missing many arguments.
Solutions exist though, as reported in The Economist, a new study by Pierre-Yves Cusset, Hélène Garner, Mohamed Harfi, Frédéric Lainé, and David Marguerit of France Strategie has managed to go around the impossibility of collecting statistics on ethnic origin in France by interrogating the origin of parents and grand-parents. As one could expect, there is widespread evidence that people with immigrant roots fare less well in France. The authors also try to identify whether things are getting better over time. This will be important to keep checking, in order to know whether the root of the problem is social inquality or ethnic discrimination.