It is a matter of convention, but has nothing to do with solvers. In the end, one usually searches for the best penalty strength alpha anyway, e.g. via cross validation.
See also this answer: https://stats.stackexchange.com/a/562494

In my case I cannot assume that my target data ist normal distributed (as required by the answer you referenced), so It would make a difference I guess.
It is true that you probably would optimize the hyperparameter again for additional data but I feel it is just inconsistent btw. the different regression methods.

Would you say that the factor of 1/n_samples is left out for ridge regression because of the fact mentioned in your referenced answer for the special case of a gaussian target Y with an orthogonal design matrix?

If we were to implement it from scratch, we would make Lasso, Elastic Net and Ridge consistent, but for backward compatibility it is what it is.
I want to stress again, that this is a purely aesthetic argument. In practice, you need to chose or find a penalty strength somehow. For instance, finding it via grid search and cross validation, the different versions of normalization would give different optimal values for alpha, but the model predictions would be the same (even same coefficients w). And that is what matters in the end.

alright - thank you, that is what I wanted to know.