anticipates his master- he is giving himself airs and graces, as you find out in Act II..;-).

He considers himself a better class of servant,IMHO, to poor plodding Thomas the coachman who has no Bath fashionable polish( but probalby has gloves if he is a coachman)

Shaking hands while wearing gloves,according to the section Etiquette for Men in the anonymous book Etiquette for Ladies and Gentlemen(1825) , published in Halifax, Yorkshire was done only with those to whom you felt socially superior. So Fag is being 'superior" in his manners to poor old Thomas.

I suppose the reason Thomas might not reply to this is that he might not notice, or might even be puzzeled by Fag's fastidiousness ;-)