No, colour is not routinely checked for when doing passports - think how many horses have the wrong colour entered in their passport. The DNA test my horses had is for parentage only (AHS passport).
Some Studs will be happy to DNA for grey for you, it depends on the individual people involved. You can ask and see what they say.
As a side note - I have seen more true Duns already DNA'd ready for sale (especially quarter horses and PRE's) than Buckskins (unless a PRE as they have both the Dun and Cream gene in the breed) so the people breeding those breeds have welcomed DNA testing for colour, which is lovely.