united-states
In U.S. law, this probably wouldn't be the basis for a civil action unless the publication was such that the author having a PhD was a necessary part of its value (e.g. an expert witness report in a court case). This would, however, be almost entirely a matter of state law, rather than federal law, so the answer might not be 100% uniform across every U.S. state and jurisdiction.
There is a long tradition of pseudonymous publications in U.S. history, going all of the way back to the Federalist Papers. Generally, this practice, or other misrepresentations about who wrote something, isn't actionable (a different analysis might apply if you were selling, e.g., an allegedly original journal purportedly written by a famous person that would be virtually worthless but for its authentic connection to the famous person, i.e. where the buyer is buying the authorship and not the content itself and only the authorship is material). While like of liability in these circumstances isn't necessarily compelled by the First Amendment to the U.S. Constitution, it is colored by that tradition.
For example, the book "Rich Dad Poor Dad" (1997) by Robert T. Kiyosaki and Sharon Lechter, is full is major misrepresentations about Kiyosaki's personal experiences which are a core part of the argument that the book makes for the personal finance advice that it gives. But this is not actionable in U.S. law.
Unlike Germany, the U.S. has almost no statutory regulation of when academic and professional titles can be used beyond common law fraud. But, common law fraud sets a high bar to claims.
Generally, if the book is the book that the title proclaims it to be (and not, for example, a knockoff of another real book with a misrepresented title and author written by a third-party's AI), a misrepresentation of the author's identity or credentials will not constitute a material misrepresentation of fact for purposes of common law fraud, which is an element of that claim.