For now we only store (and use) travel time matrices as all distances are computed after optimization with a route request to the routing engine. Storing distances is a technical step required for handling any distance-related stuff during optimization, e.g. #354 or #908.

I think both ORS and Valhalla provide distances out of the box so it would "just" be a matter of parsing them and re-using our internal matrix logic for the additional distance matrices. For (lib)OSRM, we'd have to add the required additional flags to get distances on top of travel times in the initial table request.
We'd also have to adjust the custom matrices logic to allow passing in distances matrices per profile.