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Opinion polarization is on the rise, causing concerns for the openness of public debates. Additionally, extreme opinions on different topics often show significant correlations. The dynamics leading to these polarized ideological opinions pose a challenge: How can such correlations emerge, without assuming them a priori in the individual preferences or in a preexisting social structure? Here we propose a simple model that qualitatively reproduces ideological opinion states found in survey data, even between rather unrelated, but sufficiently controversial, topics. Inspired by skew coordinate systems recently proposed in natural language processing models, we solidify these intuitions in a formalism of opinions unfolding in a multidimensional space where topics form a non-orthogonal basis. Opinions evolve according to the social interactions among the agents, which are ruled by homophily: two agents sharing similar opinions are more likely to interact. The model features phase transitions between a global consensus, opinion polarization, and ideological states. Interestingly, the ideological phase emerges by relaxing the assumption of an orthogonal basis of the topic space, i.e. if topics thematically overlap. Furthermore, we analytically and numerically show that these transitions are driven by the controversialness of the topics discussed, the more controversial the topics, the more likely are opinion to be correlated. Our findings shed light upon the mechanisms driving the emergence of ideology in the formation of opinions.