As far as I can tell (although it's always difficult reading the sales blurb and trying to work out what something actually does), the built-in media server takes advantage of the Play's "media capable" Intel Atom in order to send files to Synology's own Video Station application. There are already NAS's that can do video transcoding but only the higher end models (some of the "plus" models in Synology terms).
It would require a third-party solution to open this up I think, which I would imagine entail installing MC on the NAS. The Linux version of MC is under development but I think it is too early to comment on whether there will be a NAS-installable server version, and whether it can run under the different types of processors that are in NAS's (Arm, x86, etc). The emphasis at the moment is porting MC to desktop versions of Linux.
Apart from that, any NAS will be fine with MC for the storage of media files and using it as a file server, like an external drive. However if you want a NAS to be a media server then the only option at the moment is to use its native server, or another third-party server that is already available for NAS installation.