Interestingly, one of the most useful documents giving us insight into the original staging of Shakespeare’s plays depicts characters from one of his Roman-set plays: the so-called Peacham drawing (c.1595). This contemporary image (https://shakespearestaging.berkeley.edu/system/files/images/1595TAScene.jpg) shows a moment from ‘Titus Andronicus’, Shakespeare’s early Senecan-style Roman tragedy (the one where Titus feeds Tamora her own rapist sons cooked into a pie...). It suggests that costumes of the time combined early modern elements (the Elizabethan-clothed attendant soldiers on the left) and an attempt to evoke Roman dress (Titus in a toga-like robe in the centre). Aaron the Moor on the right would have been a white actor (like Richard Burbage, the first performer to play Othello) in a dyed lambs-wool wig and black make-up (possibly charcoal).

TLDR: Sartorial accuracy was not a priority, it appears, but Shakespeare’s company would have made some effort to convey a sense of historical era and location through costume.