Is there some sort of rule that says that there can only be two political parties.

No, but I think the issue lies with their system of government. As a presidential republic they don't have parliamentary government where parties enter coalitions to form the executive, so smaller parties have no good way to get established and share executive power and responsibility while growing.

Couple that with the voting system which also favours the trend to only two parties, and there you have the mess.

PS: They have the Green Party and the Libertarian Party, but they remain at the fringe and largely irrelevant. No seats for either of them.

It's basically a mathematical consequence of First Past the Post voting system. Until we get Ranked Choice Voting or similar, we will always only have the 2 parties. And since we need at least one of those parties to support a new voting system, I'm not holding my breath to see a sweeping change there. Happily, a couple states have embraced Ranked Choice Voting, so maybe a change can occur in time.

Does the US really not have any other political parties?

We do, but they've been institutionally ostracized since the 90's, and very few people vote for them. The two ruling parties have the backing of the billionaire class and the national news media, so likely many aren't even aware of other options to begin with.

Technically we can have any number of political parties and do. I believe I had candidates representing five parties running for my state’s auditor this last time.

However various conditions have always prevented other parties from building a competitive national presence. Things like name recognition and the vast amount of money spent in a typical election is very difficult for a new party. On the occasion when someone has reached the legislature, you can’t really do anything without a party behind you, so it’s tough to make an impact. People complain about the voting system, but it just means there’s no second place: each of the two will win different areas but very tough for a third party to break in since there’s no second place or third place.

It takes time to grow a new party. With first past the post voting there is no time to do that in. Since any new party will take voters from the party you allign more with and guarantee the party you align less with wins. And none of the 2 major parties are interested in changing to a system that would give themselfs less power. Those poor people over there are stuck with psycopatic fasistic party that is afraid of everything and everyone, and a megalomanic ignorant party that degrades into in-fighting any time they have any power.