A 2020 research paper published in the Journal of Marlowe Studies claims that Barrett Emerson is purely fictional, with some justifications of why such a character might be invented for the TV show and how he would fit in with the characterisation of Marlowe therein.
In this paper, Emerson is first mentioned as an "invented character", with parallels drawn between Marlowe's life with him and the story of Doctor Faustus:
At the very end of Episode 5, he enters the bedchamber of a fine house and lies down alongside an older man who is dying of a consumptive disease. This figure, we find out at the end of the season, is an invented character named Barrett Emerson, who was once Marlowe’s lover. [...]
Emerson believes that, like Faustus, he has sold his soul, but instead of 24 years of perfect knowledge, he receives in exchange Marlowe’s love. After Emerson dies in the next episode, Marlowe toasts his corpse with an admission that he has made a similar trade:“If your love cost me my soul, then here’s to damnation, darling. I shall see thee in hell.” Emerson’s warning, “Repent yet, and God may pity thee,” anticipates an admonition that Faustus will receive from the Good Angel—“Faustus, repent yet, God will pity thee” (2.3.12)—but Marlowe, like Faustus, refuses to repent and beg God’s forgiveness.
The contribution of Emerson to the play Doctor Faustus actually diverges from what real-life experts believe another writer may have contributed to this play, which is further evidence for Emerson being a "purely fictional" character invented to support the conception of Marlowe built up in this TV show:
Emerson has served Marlowe as a type of muse, collaborating with him on the composition of his most recent play. Modern scholars generally accept that Marlowe worked with some other playwright on Doctor Faustus, but according to editors David Bevington and Eric Rasmussen, “Marlowe wrote the serious and tragic portions” of the drama while “a collaborator took responsibility for the comic horseplay.” Consistent with Pearce’s conception of authorship as a reflection of biography, the play produced by his fictional Marlowe concerns only the “serious and tragic” story of a man who sells his soul to the devil and is damned for it, not the “comic horseplay” of the promising scholar who wastes his considerable abilities. Pearce endeavours to align his portrayal of Marlowe with the mythographic image of the tragic playwright, which requires him to ignore the play’s middle scenes and replace the anonymous comic collaborator with a purely fictional muse, Barrett Emerson, who provides Marlowe with material only for the tragic sections of the play.
It is also suggested in the TV show that Emerson painted a portrait of Marlowe as a young man, but again this diverges from the likely reality, since although the portrait does exist, and has been identified with Marlowe in "the public consciousness", there is evidence that it's unlikely to actually have been Marlowe:
The painting in Emerson’s room is recognizable as a representation of the portrait found in 1953 during construction on the Master’s lodge at Corpus Christi College, Cambridge, where Marlowe was once a student (see Figure 3). Dated 1585, the portrait identifies its subject as being twenty-one years old, and Marlowe was that age in that year. Of course, there would have been many twenty-one-year-olds at Corpus Christi at that time, and nothing in the portrait indicates that the young man was a student in the college. Moreover, J. A. Downie points out that the appearance of the sitter does not match the known facts of Marlowe’s circumstances in 1585: “As the lavishness of the costume attests, the portrait is evidently of a wealthy young man. Marlowe was a cobbler’s son, at Corpus Christi as a Parker scholar; he is therefore highly unlikely to be the subject of the controversial portrait.” Despite this contrary evidence and the lack of any positive support, “the identification of the figure in the portrait with Marlowe was simply too tempting to resist,” and this likeness has entered the public consciousness as a representation of the playwright’s dashing countenance.
So it seems that Barrett Emerson didn't actually exist, nor is he based on any real character (one can't count the other contributor to Doctor Faustus, since that real person's contribution was evidently quite different from what the fictional Emerson contributed). But Emerson's role in the TV show ties in to the way Marlowe's character is portrayed and built up there, so it makes sense that such a character would be invented.