Does the US Passport provide any kind of immunity against harassment, insult and intimidation by the border police of a destination country?
As others have pointed out, no. Harassment is generally prohibited by local laws, but of course what you consider harassment and what the local law considers harassment may not be exactly the same thing. In general, government officials everywhere enjoy a presumption that they are acting reasonably, and even when they are not they usually have an easy time arguing that their actions were reasonable. Just like a police officer in the US, they can claim that some aspect of your presentation at the border aroused their suspicions, and it is difficult for anyone to prove that they haven't made that up.
Historically, a passport was a diplomatic request from one sovereign to other sovereigns to afford legal protections to a traveler. Whether that is still the case is probably a matter of debate, but to the extent that it is true it means only that you have access to the destination country's courts. So, if you really can make a case that your treatment was illegal under that country's laws, you may be entitled to some relief. This is extremely unlikely in general, but if you want a more specific assessment of your particular case, the proper course of action is to speak with a lawyer in that country. The local embassy should be able to help you find a lawyer who speaks English if the local language is not English.
The US passport definitely does still entitle you to some protections and support offered by the US government in certain cases, but that is very different from "immunity." For example, if you are arrested or prosecuted for a crime, you have a right to consular assistance. The Department of State is very clear that their ability to assist is limited to certain specific areas.
If you think about this a moment, you'll see why it must be so. Diplomatic agreements are generally reciprocal. If there were some rule prohibiting you from being detained at the border for an hour, then such a rule would also apply to citizens of that country who seek entry to the US. Would you want everyone in that country to enjoy "immunity" from such treatment? I don't think so. Immigration officers at national borders are charged with identifying and preventing people who are entering a country illegally, and most people want them to be able to do their jobs. The question is whether they were reasonable to suspect that your passport was fake and whether their response to that suspicion was also reasonable, and, once again, the proper venue for examining those questions is the courts of that country (or perhaps an administrative review if you file a formal complaint with the country's border agency).
In a comment, you write
why does it matter which passport you carry for entrance? I was under the impression that there's an agreement which clearly defines boundaries on what they can or cannot do.
From the immigration officer's perspective, the passport is mainly significant because it determines whether you need a visa to enter, and, if so, what kind. This is sometimes, but not always, the result of a negotiated agreement between the two countries. Aside from varying visa requirements, most countries guarantee the same due process of law to all arriving visitors regardless of the country that issued their passport. (I'm ignoring diplomats here, who are indeed immune from all sorts of border formalities, but there too countries don't generally treat arriving diplomats differently depending on the country they represent.) Another significance of the passport is simply that it determines which consulate the country's authorities must notify if you are arrested.
In conclusion, the US does have the capacity to use diplomatic pressure to influence how other countries treat US citizens, but for the most part it will refrain from applying that pressure except in egregious cases. For the most part, like pretty much every other country, the US will only expect each country to treat you lawfully under that country's own laws, and, if it does not, the US will probably not do much about it unless it suits the US administration's diplomatic and political agenda.
