This video is inaccurate and extremely biased.
Firstly, "republic" in the sense that the founding fathers used the term, means "representative democracy," so it is deceptive to say that 'America is a republic, not a democracy.' A republic is a type of democracy. Even more anarchic forms of democracy, like democratic confederalism or the model of the Spanish anarchists, is technically a form of republicanism.
The way that this video defines republic is flat-out wrong. The video states that a republic is "one where the government is limited by law." That's not a republic proper. That's true only of a constitutional republic. And, that definition also fits forms of government that are clearly not republican: e.g. constitutional monarchies are limited by law, but may not have any sort of representation for the people. A constitutional monarchy is, however, NOT necessarily a republic.
The left-right spectrum originated with the French Revolution and alludes to the fact that radicals (republicans, anarchists, socialists) sat on the left in parliament, while conservatives (aristocrats, monarchists) were on the right. So, to be on the left generally meant to be a supporter of radical democratic reforms or radical economic reforms. Technically, Bastiat was on the left, even though he was a free-market libertarian.
I don't currently identify as an anarchist, but I'm quite familiar with anarchist theory in general, and this video inaccurately represents anarchism. Many anarchists (e.g. Pierre-Joseph Proudhon, Mikhail Bakunin, Peter Kropotkin, Murray Bookchin) were/are radical republicans (i.e. they advocate a form of representative or delegative democracy).
This video is riddled with inaccuracies and demonstrates a clear failure on the part of the producer/narrator to grasp basic concepts of political theory and political philosophy.