Cross validation is used to assess the validity of a particular finding, usually tangibly related to a prediction model. These kinds of findings can include results from cluster analyses, classification, or prediction models. In any case, the nature of the analysis is part of a pre-specified question of scientific interest such as "which frequencies of mRNA expression most likely originated from breast cancer biopsies versus healthy controls?". Cross validation is a very robust way to assess the validity of a prespecified prediction model that serves a specific purpose. I suspect that $k$-fold cross validation, which involves repeated model fitting, may be the origin of your confusion.
When using CV to identify an "optimal" tuning parameter, as is needed with penalized likelihood methods, there is usually a prespecified criterion that the process is set to meet. This might be minimal MSE, or maximal AUC, or minimal BIC. If you cherry pick a $\lambda$ which gives you results you are after, then you have done something worse than data dredging, I think. So, using a microarray example, if you are interested in which proteins are more expressed in cancer cases versus controls, you might prespecify a GLM LASSO to have $\lambda$ give the best BIC and markers selected in the final model are chosen as candidate proteins for further investigation. This is an example of feature selection.
"Data snooping" or, as I could call it, "Exploratory data analysis" does not deal with a pre-specified question. You kind of enumerate a number of possible, plausibly interesting results and evaluate them individually. You can perform any number of exploratory analyses and, usually, you don't worry about multiple testing. You can assess each exploratory analysis individually using cross-validation, but it does not inherently account for multiple testing when you have more than 1 exploratory analysis. Hypotheses in this setting can be quite wide and far reaching, "which factors are associated with prostate cancer?" (from which coffee drinking, vasectomy usage, etc. etc. were measured in a cohort). Significant results are seen as "hypothesis generating" and provide no confirmatory evidence.
So while both approaches are somewhat "iterative" in nature, they are entirely independent procedures. $k$-fold cross-validation is a tool for assessing uncertainty and validity of a particular set of findings which are part of a "modular" hypothesis. Data dredging, data snooping, or exploratory data analysis is meant to generate hypotheses based on a large set of possibly interesting questions addressed by a comprehensive and large dataset.