In 2013, @Donbeo asked whether there were any theoretical results supporting use of Cross Validation to choose the lasso penalty, and was scolded in the comments for asking "a pretty generic question about generalization error and empirical risk minimization." Well, I think was a good question.
(Picture from Zou's paper referenced by @Edgar in his answer)
I know it wouldn't work out well to try to estimate $\lambda$ in a frequentist maximum likelihood setting. If I had to propose why, I'd say there are problems with identifiability. But if that's true, then there must be some magical property of Cross Validation (or Empirical Risk Minimization in general) that allows one to estimate it without making any other assumptions. I would appreciate any thoughts on this.
Most of all, I'd like an explanation of what types of parameters in general is Cross-Validation more suited to than traditional inference, and some rationale as to why.
P.S. This post is an interesting read about CV as it relates to empirical bayes, but it focuses more on CV's ability to counteract model misspecification.