Isn’t much of Obama’s money coming over the internet? If so, then there is an electronic trail of names, addresses and account numbers so there is no reason to keep then anonymous.
You're right. Any contributions from individuals through ActBlue, Moveon.org, etc. are remitted along with a detailed list of the contributors, regardless of the amount.
Furthermore, while the law doesn't require disclosure for contributions under $200, if they aggregate more than $200 they are fully reportable, from dollar one.