What would be a reason to use the Root Mean Square Error (RMSE) to combine data? In this machine learning tutorial by Google they use the Root Mean Square to create a similarity measure between two shoes.
They first calculate the difference between the size of two shoes and then do the same for their price. Finally with those two values they compute Root Mean Square Error (RMSE) however they don't explain why it's appropriate to do so.
Could someone explain why would you use RMSE or in what case does it apply?

RMSE is a widely applicable and often used error function, but it is certainly not the only one or the best one for all circumstances. It has the sometimes-desired feature that bigger differences count for more, so a difference of two shoe sizes will more than double the feature-wise error compared to a difference of just one shoe size. Under this measure, two shoes that have a standardized size difference of 0.1 and a standardized price difference of 0.1 are more similar than two shoes of identical size but with a standardized price difference of 0.2.  You can contrast this with the mean absolute error (MAE), for example, which simply sums up the magnitude of differences - here, all differences count for the same weight, larger differences do not count for exponentially more error. With MAE, the previously mentioned pairs of shoes (0.1 and 0.1 difference vs. 0 and 0.2 difference) are identically "different" from one another.
The choice of error function will be domain dependent. The RMSE is a reasonable choice for many applications, and seems to be chosen in this example for convenience. Other error measures would also be justifiable for this problem, and there's nothing particularly special about this case to suggest that RMSE is inherently the best error measure to use here.
