Shakespeare's stage in many ways mimicked offstage gender norms rather than completely overturning them -- we don't have records of any performers from this era who might be understood as transfeminine in their private lives, though it's possible they existed. Women's roles were largely taken on by young male actors, who were expected to take on male roles as they aged out of beardless adolescence and into adulthood; it would have been contrary to prevailing practices for a skilled performer who had aged out of youth to continue playing women's roles out of personal preference or a more lasting identification with womanhood. Again, it's possible this happened, but we don't have a record of it, and it wouldn't be sustainable as a career-long practice. For a young man to play women was a stage in the development of male actors, from an adolescent newcomer effectively apprenticed to older performers to an established and mature company member. Some of the most successful and high-profile actors of men's parts got their start playing women; this doesn't appear to have been held against them later in life, and no known Elizabethan actor seems to have presented as a woman in private life. This trajectory of "maturing" from women's parts to men's could have been a real pain for performers who for whatever reason preferred roles like Cleopatra and Juliet to roles like Othello and Lear.

In many respects English drama of this period enforced otherwise-normative standards of what women looked and sounded like, capitalizing on the physical qualities of prepubescent male-assigned performers such as higher voices and scant facial hair. Plausible performance of femininity was assisted to an extent by stage conventions of dress and gesture. 20th and 21st century productions of Shakespeare often lean harder into the deliberately gender-transgressive or campy aspects of cross-gender casting, while discarding the expectation that there's a time limit on how long male actors will play women or that performers should seek to mimic a certain body type; for instance, Mark Rylance playing Olivia in Twelfth Night at the Globe in 2013 would have had a good thirty years on the average Elizabethan actor who would have originated the role.

Audience members might well have responded anyway to the depictions of gender transgression onstage -- crossdressing and recursive crossdressing, nominally heterosexual love stories played out by two people of the same sex, supernatural characters who defy gender categorization, and stage depictions of real-life gender transgressors like the protagonist of The Roaring Girl. This gender play could exist on multiple levels -- "boy actors playing women dressed as men dressed as women" is the classic example. Certainly, critics of the theater responded strongly to transgressions like boy actors playing women, as well as actors of all ages playing characters of higher social class -- two critiques that hinged on seeing through the stage spectacle to the "real" identity of players. Critics and polemicists likely wouldn't have responded warmly to someone assuming the clothes of another gender than the one they were assigned at birth, but it was in part the very public nature of the theater that exposed performers to criticism, and it's hard to say from this what criticism an individual who chose to live as another gender might face outside the theatrical context.

Would some Elizabethans have identified themselves as transgender if that category and concept were available to them? Probably. Would some of them have identified that category with the kind of gender play depicted onstage and in contemporary literature? Probably, but it's hard to say with so little data available about people who exhibited trans-like behaviors in this period and their lives. It's hard to say where else gender-nonconforming individuals were congregating in this era, in part because records disproportionately reflect brushes with the law and in part because modern transgender identities don't map too well on Elizabethan and Jacobean understandings of gender and dress. Elizabethan polemics against gender transgression in clothing and manners don't necessarily reflect a cross-gender identification in their targets, but rather an anxiety about the transgression of gender boundaries on the part of the polemicist. Mary Frith (the real-life inspiration for the titular character of The Roaring Girl) was understood to be a woman but wore breeches and doublets, ostentatiously smoking tobacco in an era where smoking was widely regarded as a man's pastime. By 17th century standards, she ostentatiously transgressed gender norms about as publicly and openly as a person could; however, this transgression had its own distinct connotations that don't necessarily have parallels in modern discourses of gender. (The notion that DFAB people wear men's clothes out of heterosexual promiscuity seems to have fallen by the wayside, for instance.) Mary's life and semi-mythical exploits as depicted onstage can tell us something about contemporary Jacobean understandings of the body, gender performance, and societal disruption, even if we can't necessarily fit her experience neatly into modern terms.

You might be interested in these past threads:

• "In Shakespeare's day, were female roles really played exclusively by teenage boys?" answered by /u/amandycat

• "How could a 15-year-old boy have pulled off the role of Lady Macbeth?" answered by /u/cdesmoulins