The “political” aspect of communism stems directly from the desire to radically alter the economic system. It is not tied, however, to the particular political order.

Coming from the same very Wikipedia article you cite on communism:

Communists often seek a voluntary state of self-governance but disagree on the means to this end. This reflects a distinction between a libertarian socialist approach of communization, revolutionary spontaneity, and workers’ self-management, and an authoritarian socialist, vanguardist, or party-driven approach to establish a socialist state, which is expected to wither away.

So, communism, just as capitalism and socialism, can be combined with all sorts of governance types. It can be authoritarian (and so can be capitalism - look at fascism to see an example), and it can be democratic (early Soviets) or even libertarian (anarcho-communism). You can build a totalitarian communist hellhole, and a totalitarian capitalist one; same in reverse.

Now, an argument can actually be made that capitalism is inherently undemocratic. As your ability to exercise rights is heavily tied to your wealth (think of regular worker suing a billionaire, or all the lobbying, or corruption scandals involving the wealthiest and the way they slip out of them like nothing ever happened), people can be and commonly are silenced. Moreover, if you have money, nothing stops you from financing the media to translate your message. This way, important political messages are drowned in favor of what the rich want to translate, and certain (rather corrupt) voices are heavily amplified over others.

By extension, liberalism, even in the most ideal of its forms, is deeply flawed when it comes to a true democracy.

Finally, most communists (including Marx, since you mention him) realize that the communist society is at least very far off from the current state of affairs. This is why socialism exists as a transitory state, an economic system that grants a lot of benefits of communism (worker’s rights, a social state, socially owned industry) while keeping the monetary incentives in the economy. The absolute majority of communists support this transition and welcome a socialist state.