Linear regression uses a measure of an "error" for every data point. Visually, the error is the vertical difference between a data point and the line/plane of linear regression. In contrast, PCA measures the distance from the data point along the line perpendicular to the PCA axis. The PCA distance is also known as a "projection".

There is something known as orthogonal regression (total least squares) which uses the same measure as PCA. Unfortunately it doesn't work well across incompatible variables.