One does not lose legal rights by becoming a government official, so POTUS retains the right to sue for defamation, hold copyright, sue for trespass or breach of contract, and so on. The standards for defamation change when one becomes a "public figure" (you have to show "actual malice"), but this is much broader than being a government official. Anything that is a "work of the US government" is not protected by copyright, so presidential decrees, as government works, are not protected by copyright.
I do not know of any state where one legally loses publicity rights as a function of being famous, or being an elected official. California Civil Code §3344 spells out the right of publicity in that state, which says that anyone who
knowingly uses another's name, voice, signature, photograph, or
likeness, in any manner, on or in products, merchandise, or goods, or
for purposes of advertising or selling, or soliciting purchases of,
products, merchandise, goods or services, without such person's prior
consent
can get sued. However, there is a "fair use" escape clause:
For purposes of this section, a use of a name, voice, signature,
photograph, or likeness in connection with any news, public affairs,
or sports broadcast or account, or any political campaign, shall not
constitute a use for which consent is required under subdivision (a).
The law doesn't say exactly what constitutes a "political campaign" or "public affairs broadcast or account", but since politicians get caricatured in the papers all the time, with no requirement for consent, it is highly likely that the use you point to would be found to be part of a "political campaign" or "public affairs account".
Additionally, under the First Amendment, you can criticize a government official, and that right is not limited to just critical words. It is obvious that the things on sale are basically criticism of POTUS, and you can't use the law to suppress such criticism. Accordingly, one could also criticize Tom Cruise (not a government official) using his likeness on such an object. However, one cannot exploit his image to sell perfume.
