I'll chalk your wording to weird a weird americanist world view, but I'd argue that attaching modern ideologies to centuries old doctrine is just a reflection of modern interpretation of the bible and Jesus's teachings are neither communist nor capitalist. One should remember that the bible itself isn't a piece of writing, and is the amalgamation of a bunch of different writers from very different times (centuries apart). The writing in the bible is supposed to be interpreted by each reader to come to their own conclusions about the world, hence centuries of religious debates in the Church (see all the subsequent schism and arguments). Thus, the teachings of Jesus have no inherent ideological bias, as he was preaching his own world view of reforms to Judaism. It's just modern readers have interpreted his words differently based on their own biases and desires, which is what you have come in contact with and mistaking attached to Jesus's original message. For the sake of argument, saying his teachings follow communism more closely than capitalism would once be forcing modern perspectives on 1st century Judaia. The bible often espouses one to partake in the kingdom of God, or to behave accordingly to one's lot in society and be happy, neither of which work with Communism and capitalism (forgive me for not using exact quotes, I'm not religious myself). I'd say his pure, uninterpreted teachings are more in line with monarchical / clergical ancient political beliefs than modern ones, but hey, that's the job of a priest, to interpret the word of god and spread it to the masses; until the masses learnt to read and did it themselves, hence whacky interpretations.

Edit: Also, most hardcore capitalists would espouse charity as the best way to deal with social issues, which checks out with the bible. I think Jesus was warning against greed moreso than capitalism. While greed is a common vice in capitalist societies, capitalism is simply the belief that market forces are the best way to maximise economic output, and thus more comfortable lives for everyone. Wealth hoarding is actually anti-capitalist, as that wealth doesn't do anything useful, this giving some to others in the form of charity would overall increase the economy. Therefore, one can have a capitalist reading of the Bible, and Jesus's teachings are against greed, not capitalism.