When registering a copyright under US law, the claimant is the person who claims to own the copyright at the time of registration. This need not be the author. It could be a person or firm that the author had sold or given the copyright to, or the author's heir. It could be the employer of the person who created the work if a valid work-made-for-hire agreement applies to the work. But it is often the author.
Everything stated on an application for US copyright registration becomes a matter of public record, and may be posted on the Copyright office website or disclosed in Copyright Office searches.
The OP writes:
I understand that by leaving blank the space asking for an author's (legal) name but indicating the author's pseudonym that the two names won't be linked and thus the correlation will not be made publicly available online or otherwise.
That is not correct, or at least it is too strong. If the author's legal name is not listed on the registration application, then it will not be included as such on public copies of the registration, but if the author's name comes to the attention of the Copyright Office in some other way, it may be included in searchable Copyright Office records, depending on the exact details. The Office does not guarantee anonymity.
In any case, the name of the copyright owner will be public, and people may assume that this is the true name of the author, correctly or incorrectly.
It is very hard to publish a work, register a copyright on it, and enforce that copyright, collecting publication fees or royalties, while remaining strictly anonymous. Professional authors, some assisted by lawyers, have in some cases tried to do this and failed. The case of "James Triptree Jr" later revealed to be Alice Sheldon, and the case of "Sam Holt", revealed on publication to be Donald Westlake, comes to mind.
This answer deals with US copyright law. I suspect that the law in many other countries is similar, but i do not positively assert this.