I think we first need to unpack the historiographical methodologies underlying Bloom's claim before we can turn to your question regarding reception and interpretation on the part of sixteenth-century London theatre-goers. So, let me begin by examining some of the reviews from academic journals. Alvin G. Burstein's review (published in Soundings: An Interdisciplinary Journal vol. 83, no. 2, pp. 493-499) of Bloom's book (that I assume you're citing), Shakespeare: The Invention of the Human. Burstein states that Bloom is building upon the assertion made by Hegel that Shakespeare introduces characters into the theatrical canon that are "free artists of themselves" and, thus, Bloom finds that Shakespeare unlocks a modernist aesthetic in which characters "seem more vivid and real than the plays that they are in." Burstein continues in his review to explain that Bloom's use of this Hegelian concept is left without explanation and "lacks the index and footnotes that would enable one to find what Hegel had in mind." But, Burstein is not without praise for Bloom's efforts, stating that Bloom's "claim that these Shakespeare characters display acute self-consciousness and in that sense are choosing who to be seems well grounded." Furthermore, he is also in agreement that Shakespeare is plausibly/likely the prime originator of this aesthetic phenomenon, at least in the so-called Western Canon. But ultimately, Burstein finds Bloom's premise--which is at the core of your question--to be incredibly radical: "Bloom's central claim...is that the Bard's plays changed human nature, that Shakespeare invented, not just Hamlet and Falstaff, but us as well." And that "Bloom deals with [this claim] allusively, not definitively." Phyllis Rackin, in her review (published in The Sixteenth Century Journal vol. 31 no. 2, pp. 591-592) puts these historiographical issues into sharp focus, explaining that "Bloom's Shakespeare is the Immortal Bard who summons us to a return to tradition and presents us with "universal" truths and values independent of historical circumstances." It is critical to examine Bloom's historiography because his perspective takes from the teleological argument that the aesthetics of character representation throughout history were naturally awaiting the arrival of a "Shakespeare" that would come to unlock a most human picture within a mimetic form that could then be propagated widely through the phenomenon of popularity. That it happened to be a European of the sixteenth century is of no surprise to Bloom as that would explain the primacy of the so-called Western Canon throughout world literature and theatre.

Now, I'm not here to just attack Bloom and the foundations of his scholarship. Your question is incredibly valid! But, I must adjust a little from what you're asking because the idea of a three-dimensional character is tied to the education and experiences of modern individuals and our reception and interpretation of character mimesis. What the sixteenth-century theatre-goer of London would have understood as a true, realistic, or "human" representation of a character must be examined "not simply [within] the encoded production and text, but also the broader cultural frames that themselves situate these media or theatrical performances" (Tulloch, 87). Bloom is making a very HUGE claim. I think it best to examine the reception of character in more microcosmal doses. For example, in an analysis of the affectual element of surprise that Aristotle identifies within the genre of tragedy--as it might have been received by London audiences in 1609--Raphael Lyne explains that Ben Jonson's play, Epicoene, contains a few plot twists that come in the form of character reveals near the end of the play. For Lyne, "a full assessment of the surprise value of this scene would have to take into account the fact that many people seeing or reading Epicoene for the first time already know that a shock awaits them" (91). How so? Well Jonson leaves what Lyne calls "pseudo-clues" for either the reader or the viewer that likely would have registered with an audience--perhaps not consciously at the moment--that was culturally situated within the contemporary aesthetic framings of that day. In other words, if you or I attended what we expected to be a scary movie we would, upon hearing the music reach a fever pitch, find our bodies to be tense and our minds to be prepared for something terrible to happen or scary to jump out--even if we're not immediately aware of those things happening to us.

So, what does that have to do with the reception of a three-dimensional character from Shakespeare's plays in the sixteenth century? Well, the fact is that we have to take each character, play, and context separately. To say that Shakespeare invented the modern human is to completely ignore the way in which the larger conditions of politics, society, conflict, economy, culture, race, gender, religion, etc. play into the minutiae of audience reception. In particular, the audiences of sixteenth-century London plays were incredibly diverse and could include aristocrats, courtiers, ambassadors, merchants, men, women, children, foreigners, etc. The characters that Shakespeare presents on that particular stage were, like his contemporaries, an attempt to entertain those audiences by presenting characters that were rooted in the traditions of English theatrical practice going back to the medieval mystery and pageant plays and also informed by the growing educational reforms that brought wider knowledge and appropriation of Classical Greco-Roman dramatic forms. That Shakespeare would become the great Bard of Early Modern character creation was not immediately aware to his audiences--they simply enjoyed the experience of seeing those characters on stage. The ways in which this enjoyment was expressed and its root causes are discussed and debated through the many critical lenses of scholarly work today. Later playwrights like Jonson, Fletcher, Jones, and others would build upon what made Shakespeare successful in his own day and would adjust to the attitudes and influences working upon the theatre and its audiences in their own time. In fact, this would lead to why Shakespeare becomes perceived as such a great playwright by much later literary greats like that of John Dryden and by some of the first celebrity actors like David Garrick: they found that they could employ Shakespeare as a symbol of theatrical greatness that would be readily received by the audiences of their times and could bring them greater acclaim. Indeed, that is the tradition that Bloom is actually contributing to here.

Bibliography:

Lyne, Raphael. "English Guarini: Recognition and Reception." In The Yearbook of English Studies vol. 36, no. 1 (2006): 90-102.

The Cambridge Companion to Shakespeare on Stage. Edited by Stanley Wells and Sarah Stanton. (Cambridge: Cambridge University Press, 2002).

Tulloch, John. "Approaching Theatre Audiences: Active School Students and Commoditised High Culture." In Contemporary Theatre Review vol. 10, no. 2 (2000): 85-104.

EDIT: Thanks for the gold!