The play in the current Arya storyline in Game Of Thrones is very clearly intended as a sort of reference to Shakespearean history plays (and, I suspect, something of a meta-commentary on the changes the TV show has made to the events portrayed in George R.R. Martin's books). Like Shakespeare's history plays, the play in the Arya storyline very obviously portrays historical characters and situations differently to 'real' events in the show, and the differences between real life and portrayal occur both for purposes of entertainment - Kevin Eldon's face-pulling buffoon of a Ned Stark is played for laughs in a way that Sean Bean's Ned Stark is not - and in ways which show a bias in favour of the current Westerosi monarchy (/u/texpeare elaborates on the accuracy of Shakespeare's history plays here).

And Shakespeare's plays are not medieval, as archaic as their language and form might sometimes seem to bored teenagers in English classes in high school! Shakespeare was a working playwright when Galileo discovered the moons of Jupiter and the Jamestown colony was established in North America. Similarly, as /u/GeeJo points out here, Game of Thrones features a mishmash of technologies, ways of thinking, events, characters, and cultural institutions, from a variety of different places and times in European history, some of which are from later than the medieval period. Given that mishmash, the presence of a cod-Shakespearean play in Game of Thrones isn't particularly out of the ordinary.

Thomas Leahey's textbook A History of Psychology: Main Currents in Psychological Thought (despite being exactly the history of psychology the title suggests) briefly focuses on medieval plays, largely because medieval plays can bear insights into how people at the time interpreted human behaviour. Leahey points at three basic categories of medieval plays: a) plays depicting the lives of saints and martyrs such as Diocletian written by the tenth century Saxon nun Hrotsvitha, b) the passion plays (i.e., plays telling the Easter narrative) such as The Betrayal Of Christ performed by the Baker's Guild of Chester, England in the 14th or 15th centuries, and c) morality plays, sometimes called psychomachia, such as Tide Tarrieth No Man from 1576 (which is quite late for what is meant to be a medieval play, but Leahey paints this as the continuation of a long tradition). The psychomachia externalised internal conflicts, and are the source of the stereotypical image of the man with the devil on one shoulder and the angel on the other.

Each of these categories of plays are all clearly deeply suffused with Christian beliefs, settings, and ways of thinking, in a way that Shakespeare and the play in Game Of Thrones are not. According to Leahey, medieval plays also demonstrate that "reality and all-in-your-mind were one and the same to the Neoplatonic, medieval mind...high-born and low all thought of people as coming in types based on social status and having minds that worked according to rules appropriate to each type" (p. 102-103). In contrast, some of Shakespeare's plays, argues Leahey, evoke morality plays - Iago works on the level of the devil on Othello's shoulder, for example - but there's a distinctive individuality present in the characters in Shakespeare that is absent from earlier morality plays.

(Disclaimer: I wouldn't be surprised if the picture painted by Leahey that I've sketched here is a bit simplistic - he's a historian of psychology rather than medieval theatre, after all - and so I'm very curious about what more knowledgeable scholars would have to add or subtract from this explanation.)