But... aren't they? It's one share one vote, not one employee one vote, but I think that still qualifies as democratic. Then, there are elected board members (the representatives) who hire the CEO (the prime minister?), etc.
That wouldn't fit most definitions of democracy (which would be variations of "every person gets a single equal vote"). Most companies are more of an oligarchy.
Neither is the US. The US is a democratic republic: we vote in congress, the senate, and the president who then make decisions for us on our behalf. Public companies are similar: shareholders vote in board members, who then elect a CEO for the company on our behalf.
That is assuming that there's only one share class. Companies often have different classes of membership/shareholder which have different voting powers (or none).