In the US, state and local legislatures can choose to be bicameral, unicameral or something else. As of today, all state legislatures are bicameral, except in Nebraska, which is unicameral.
Most local governments are unicameral.
The US Congress has no structural flexibility. It must be bicameral, as mandated by the Constitution.
Why are most state legislatures bicameral?
It appears to be a vestige of colonial governments:
Before the Revolution, most individual colonies had followed the bicameral model in their governments, with an upper house, generally called the council, representing the interests of the proprietor or the Crown, and a lower house, generally called the assembly, representing the settlers.
Why are most local legislatures unicameral?
On the local level, bicameral legislatures were most common until the reform movement at the turn of the twentieth century made unicameral city councils the norm by the 1930s.
While distrust of the masses and the "need" for a propertied elite to guide them has clearly vanished as an argument for a bicameral system, this is not reflected in the set-up of the legislatures at the state or federal levels.
Why is the US Congress bicameral?
The answer is pretty well summarized in the post referenced in the question:
Here's an excerpt from the accepted answer:
The U.S. Congress didn't actually start out bicameral; the original Articles of Confederation (the pre-Constitution) had a single house. The final Constitution split Congress into the House and Senate as a compromise between the large states (who naturally wanted representation to be tied to population) and the small states (who didn't want to get outvoted all day long by the larger states). Rather than pick one, they made two houses, one with representation by population and one with equal representation for each state.
I'll add this from another source, which is a bit more detailed:
The first organizational law of the United States, the Articles of Confederation of 1781, prescribed a unicameral Congress.
This changed when delegates met at the Constitutional Convention in the summer of 1787. They adopted Edmund Randolph's plan for a three-branch government and a bicameral legislature based on population after a weeklong debate dismissing an alternative proposal by William Paterson of New Jersey for a unicameral legislature in which each state would have one vote.
Paterson's plan had been supported by the smaller states, which feared that a legislature apportioned according to population would lead to dominance by the larger states.
Distrusting democracy, many of the delegates were afraid that in a single house only the members' "virtue and good sense" would prevent legislative despotism. This seemed an inadequate check. The only possible restraint was to divide the legislative authority within itself.
Ideas about an upper house, where men of property and great wisdom would take care of issues that were beyond the grasp of the masses, merged with the need to find a compromise that would provide the smaller states with a satisfactory input in the legislative process.
Accordingly the convention compromised to propose a House of Representatives, in which members were apportioned among the states according to population, and a Senate, in which each state had an equal vote.
Side note: Interesting to read how the founding fathers created the US Congress, in part, based on their distrust of democracy:
Distrusting democracy, many of the delegates were afraid that in a single house only the members' "virtue and good sense" would prevent legislative despotism.
That issue ties into this post: When is a democratic vote actually the wrong tool?