Why RidgeClassifier can be significantly faster than LogisticRegression with a high number of classes? In Scikit document, we can find this statement

The RidgeClassifier can be significantly faster than e.g. LogisticRegression with a high number of classes because it can compute the projection matrix $(1/(X^TX))X^T$ only once.

Whys is this so? Can it be explained intuitively ?
 A: This could be for a number of reasons.  I suspect it is because the logistic regression classifier is trained using Iteratively Re-weighted Least Squares (IRWLS), which requires the repeated solution of a system of linear equations that depends on the current output of the model.  For details, see Minka (2003), sections 1 and 2.  The ridge classifier, on the other hand only requires the solution of a single set of linear equations (of the same size).  IRWLS generally takes ten or so iterations to converge, and so is about ten times slower.
However, the wording of the question suggests that the ridge classifier is using the same regularisation parameter for each class, which is not something I would necessarily recommend, as it is suggesting that the matrix that is inverted in solving this system of linear equations (which depends on the ridge parameter) is the same for all classes, which would be a further saving.
Having said which, I would use multi-nomial logistic regression, which is much slower as it requires the repeated solution of a much larger set of linear equations (but can give a better classifier).
On the other hand, logistic regression methods can also be fitted using Bohning's method, which uses a fixed Hessian, which would mean that you could use the same inverse hessian repeatedly.  This would make it nearly as fast as the RidgeClassifier, regardless of the number of classes, as the slowest part is inverting the Hessian (although in practice you would use something like the Cholesky decomposition rather than actually inverting it).  See section 5 of Minka (2003).
One thing to be aware of though, is that the logistic regression model gives you estimates of class membership, whereas the ridge classifier does not.  This can be a big advantage in some practical applications, particularly if the misclassification costs (or equivalently the operational class frequencies) or if a reject option is needed, as a probabilistic classifier can deal with those things without needing to retrain the classifier.  I like the ridge classifier, as it is fast, but don't use it where probabilities are needed by the application.
Thomas P. Minka. A comparison of numerical optimizers for logistic regression, October 22, 2003 (revised Mar 26, 2007) (pdf)
A: 
The Ridge regressor has a classifier variant: RidgeClassifier. This classifier first converts binary targets to {-1, 1} and then treats the problem as a regression task, optimizing the same objective as above.

https://scikit-learn.org/stable/modules/linear_model.html#classification
So it's a linear model that has closed-form solution. Logistic regression is linear in parameters, but this is followed by logistic transformation, so we need an optimization algorithm to fit it. So it is faster for the same reason linear regression is faster than logistic regression.
A: Having a Hessian that must be updated every time the guess at parameter estimates $\beta$ changes is a necessary requirement for achieving excellent calibration (i.e., the weighted in iteratively reweighted least squares).  Multinomial logistic regression is built for prediction, and I suspect that your stated goal of classification would be better replaced with a goal of prediction, i.e., getting predicted probabilities of class membership.  Using ordinary Newton-Raphson iteration to solve for the maximum likelihood estimate of $\beta$ converges very fast, often in only 5 iterations (5 passes at the data).  So I'm not sure how much it is worth to develop shortcuts.
