Your brother is partially right.  It all depends on the type of charity.  Generally, all 501(c)(3) charities that file a tax information return are required to provide a list on Schedule B of any person who contributed (directly or indirectly) $5000 or more in cash, securities or any other type of property during the year.  The Schedule requires the name and address of the donor and a description of the property given.

For private foundations that file the Form 990-PF, the Schedule is open for public inspection, including the names and addresses of the contributors.  For public charities that file Form 990 or Form 990-EZ, the names and addresses may be withheld, but the rest of the information must generally be available.

The ability to keep contributions anonymous is one of the reasons some donors choose to establish a donor advised fund at a public charity rather than create their own private foundation.  (See Ready Reference Page:  “Donor Advised Funds Still Compare Well With Private Foundations.”)