In lots of ways, I would argue. Shakespeare's plays are often talked about as being universal, because, though they are from 400 years ago, they deal with themes and problems that come up in everyone's lives.
Viola, for example, the central character, has to deal with (what she thinks, anyway, has happened) the death of her brother at sea - and, at the end, discovers that he is still alive and reunited with him. Everyone will experience the death of a loved one at some point in their lives, and the wonder and beauty of being reunited with a loved one who you supposed to be dead is something that can strike a chord with anyone who has lost someone. These are universal concerns.
Twelfth Night thinks extensively about death: Olivia is mourning the loss of her brother and father, Viola supposes Sebastian to be killed in the shipwreck (and vice versa) and Feste is constantly making references or remarks to death (his song, 'Come Away Death' imagines what it would be like to be buried).
I'll give you two more examples of Shakespeare's universality. The basic premise of the Viola plot is that she is head over heels in love with Orsino, but cannot tell him, because she is disguised as a boy. This leads to her, late one night, saying things to him which - though he has no idea about it - are actually about her:
Surely everyone experiences the pain of loving someone who - for whatever reason - they cannot tell. Again, it is not the situations themselves but their emotional implications that are universal.
One more: think about Malvolio. He's serious, self-important, pompous, malicious and enjoys nothing more than stopping people's fun. Everyone knows someone like that (I bet you've just thought of one). And moreover, everyone takes some delight in the idea of them being humiliated; exactly as the yellow stockings plot does to Malvolio.
So - in short - you can find analogues in your own life of the situations and emotional relationships in Shakespeare's play which should help to explain why it is that Twelfth Night is still one of hte world's most popular plays after 400 years.
Hope that helps!