I can see the hesitation in wanting to provide that to an unknown lady. And you should never do anything which makes you uncomfortable. This is supposed to be an arena for alleviating stress.....not creating more!
However, she has provided alternate and more anonymous ways of screening.
I suppose reputation as well as the type of lady a gentleman sees plays a role in how much information he provides on initial contact. More than half of my initial emails from potential suitors contain all the information she is requesting.......and more! Full name, company, position in company, main line number, direct work phone number, personal cell number and multiple references from recently visited established ladies are not at all unusual in an initial contact.
I don't request that information, but it tends to be provided very easily and openly. So I don't think she will have any problems getting it from potential suitors either.
Ironically, it seems that the more prominent and 'successful' gentlemen are the ones who are most open, and the those who have seemingly little to lose if they are found out are the most guarded. Perhaps because most highly successful people have incredible gut instincts and an exceptional ability to read people. They simply wouldn't contact a person that they didn't believe was trustworthy and respectful of their discretion.
With all of that said, and with the new C-36, I would be leery about providing too much personal information to somebody who didn't have an established and sane reputation.