Random forests use bagging (bagging is just a contraction of "bootstrap aggregation," see Elements of Statistical Learning Section 8.7). Bagging draws a bootstrap sample of the data (randomly select a new sample with replacement from the existing data), and the results of these random samples are aggregated (because the trees' predictions are averaged).
The boosting implementations that I'm familiar (e.g. xgboost) will also support random subsampling of columns. But your guess is correct, in the sense that column subsampling is really incidental to the boosting procedure itself. Boosting is about estimator $T+1$ "fixing" the errors of the previous $T$ estimators.