As far as I know, the Arabian horse registry has no rules against showing a horse based on what color it turns out. Now, an Arabian might not show very well if they are an odd color, especially in a halter class, because it is not considered ideal.
As fasr as grey and brindle.... if he is a chimeric brindle, I am not sure how the combination would play out.....perhaps one or both "twins" that he is made of could have been grey, and since the brindling is the striped combination of the colors of those twins, his stripes might be greying at different rates.... or, one set of stripes might grey and the other remain chestnut (or whatever it is.) OR, perhaps the command from one set of DNA trumps the other DNA, and will grey him overall.
But to me he DOES look like he is greying. Most brindles who are non--grey look like lighter and darker shades of whatever color without the white hairs--
Here is a chestnut brindle as a foal and as a 2 year old--
Here is a brindle who is either red dun or chestnut (had seen him called both) as a foal and as an adult--
Here is a bay brindled with white-- but the bay is not lighteneing or changing at all. Makes me wonder if he is chimeric, and one twin was bay, the other grey, or one bay and one roan, or....? Or maybe his brindling is non-chimeric, but more like roan gone crazy, LOL....