OBoE is a manually moderated booking system. As an attendee you fill in your details and press a button to ask the organisers if you can come to the event. Also you 'offer' them money (by cheque or PayPal). The organisers check there are places left and that your details and the money are as needed before they accept your booking.
After you have made a Booking Request and Paid Online using PayPal or such, the organisers' administrator still has to do stuff on their side to acknowledge that the payment has arrived, that it is for the right things and that there are still places left and all the other things that they need to check.
If you are not paying online then the booking will not show as paid until they process your cheque or other transaction. The button offering Online Payment may still be displayed.
Only once they have done all this will the booking show as 'paid'.
The event organisers may well accept your booking request before checking for payment. That's down to them
A Booking Identity is an alias that allows a registered user to make several separate and extra bookings for the same event for themself and for other people. Such as their mates who don't have internet access.
You can add them in your Profile. Once you have added an alias, don't re-name it for someone else, certainly at least not until the event is over.
Event organisers don't get to see what you called your Identities. They can tell that you have used one but would only refer to it by number if they had to contact you with a problem. Which seems what you would want?
The organisers need your full, actual, real name for legal purposes like insurance. It's not an issue of whether you care if you are covered by their insurance, it's about the organisers being insured at all. Why you don't want to give them your name? Especially if you are going to pay online or by cheque...
Many people don't use their full name in day-to-day affairs. This is just the name you normally use.
Likewise you may have a nickname that a lot of people know you by. But only some people.