The OP seemed to be asking the question in good faith, with the intent of learning more about the topic. So I think the post warrants good-faith replies.
As others have said, being non-binary isn't a sexuality; the situation you're describing it's a closer equivalent to putting up a "women only" or "men only" audition notice.
I see what you're asking regarding "type" - and I think for that reason, you'll sometimes see casting notices for non-binary people with a little more specificity. For example, some might use language like "masc," "femme" or "androgynous" if it's important for the role. Or in some cases, they might specify that a non-binary character is AMAB or AFAB (Assigned Male/Female At Birth)
Or for a musical, they might say that a role is a "tenor" or "soprano" simply to clarify the abilities needed to play the role - although even this might be seen as controversial, because some have pointed out that this might be used as an industry-wide excuse not to cast trans people, when in fact they could simply change the keys for the actors. I saw someone online recently point out "if they can lower the keys for Sutton Foster, they can lower the keys for a trans person"
But I think many people are also just looking to cast shows more gender diversity. And sometimes a characters' gender assigned at birth, or specific gender presentation, aren't important to the character's story. It's just important that they're non-binary. And at that point, it basically just becomes like a normal audition process; who's talented? Who has the energy/abilities you're looking for in the role? Who seems good to work with? Who can convey the personality traits of the character most believably?
And for the record, @dramamama, I don't think it's worth being concerned about people lying about being trans or non-binary. This isn't really a thing that happens. I'm sure there are some A-holes who have done it, but buy-and-large it's not a problem that's widespread enough to use it as an excuse to cast doubt on people who say they're non-binary. This is the kind of mindset that got JK Rowling in trouble.
Updated On: 2/15/22 at 09:44 AM