I've read that multicollinearity is one of the main assumptions of multivariate linear regression - Multicollinearity occurs when the independent variables are too highly correlated with each other.
However, when learning linear regression, one of the key topics is the idea of introducing interaction terms into the model to model the interaction effect which is when the effect of an independent variable on a dependent variable changes, depending on the value(s) of one or more other independent variables.
Aren't these two statements contradictory? If there were really interactions between $X_1$ and $X_2$ in the model $y = \beta_0 + \beta_1*X_1 + \beta_2*X_2$ surely we should remove either $X_1$ or $X_2$ so that the independent variables in the regression model are no longer correlated thus the multicollinearity assumption holds. Putting in the interaction terms seems to ignore this assumption and rather introduce a further term to further complicate it.
From a modeling standpoint this makes sense, but doesn't the mathematics breakdown if we do this?