Shakespeare's plays do tend to have a timeless quality to them, as can be seen by their enduring presence in dramatic circles, all the way to the present. They've often had their settings updated to fit different time periods, including contemporary society, and this is not by accident. Shakespeare does tend to focus heavily on characterization, and universal themes of the human condition, and for all that society has changed over the centuries since Shakespeare's day, people remain people, with many of the same concerns and shared experiences.
We see this in the example of Twelfth Night, which tackles several universal themes. It is ultimately a story about love (as well as longing), but that theme extends beyond the realm of romantic love. We see bonds of friendship, between Antonio and Sebastian, to such a point that Antonio will risk his own safety, following Sebastian into Illyria, where he is surrounded by enemies. We can observe its plot of mistaken identities, with Viola impersonating a man in order to enter the inner circle of the Duke. It depicts romantic frustration, with Viola pining after Orsino, Orsino pining after Olivia, and Olivia pining after the disguised Viola, as well as the internalized conflict this situation creates, with Viola torn between her duty and loyalty to the Duke, and her own romantic desires. In these sorts of qualities, Twelfth Night presents a vivid depiction of humanity, and in this it transcends any particular time period and culture, including the one in which it was created.