Typically, the regression assumptions are:
1) mean error of zero
2) conditional homoskedasticity
3) error independence
4) normality of the error distribution
I've done some econometrics course work so I am aware how the Gauss-Markov items mention things a bit differently and add two assumptions.
Technically, absence of perfect collinearity isn't a regression assumption, but it assures unique estimates and works better for the matrix algebra because perfect collinearity would cause redundancy in the matrix and makes the matrix non-invertible (singular), which means the determinant is zero (you can't find unique solutions-- many values for the coefficient estimates could work). Avoiding perfect collinearity allows for the case of unique solutions (coefficient estimates).
If you want to make inferences on the estimated slope coefficients, multicollinearity (at a problematic level) can cause inappropriate and misguided inferences such as concluding the wrong magnitude, wrong direction, or both regarding a partial relationship of some Xi with Y. Multicollinearity, by it's nature can introduce instability in the regression parameter estimates, but the model is still unbiased and consistent, all else constant.
If you want to use a model for predictions, though, multicollinearity is not as much of a concern, generally speaking. This is because the predictions remain unbiased (as alluded to above).