This is an issue to report the step to actually take over the work of @marctorsoc in #23183 and split the PR into smaller one to facilitate the review process.

The intend is to make the zero_division parameter consistent across different metrics in scikit-learn. In this regards, we have the following TODO list:

• Introduce the zero_division parameter to the accuracy_score function when y_true and y_pred are empty.
  • ENH Add zero_division parameter for accuracy_score #29213
• Introduce the zero_division parameter to the class_likelihood_ratios and remove raise_warning.
  • MNT remove default behaviour deprecation from class_likelihood_ratios #31331
• Introduce the zero_division parameter to the cohen_kappa_score function
  • ENH Add zero_division param to cohen_kappa_score #29210
• Introduce the zero_division parameter to the matthew_corr_coeff function
  • ENH add zero_division=nan for classification metrics #23183
  • FEA add zero_division to matthews_corrcoef #28509
• Open a PR to make sure the empty input lead to np.nan in classification_report function. classification_report should raise an error instead: see Make zero_division parameter consistent in the different metric #29048 (comment)

All those items have been addressed in #23183 and can be extracted in individual PRs. The changelog presenting the changes should acknowledge @marctorsoc.

In addition, we should investigate #27047 and check if we should add the zero_division parameter to the precision_recall_curve and roc_curve as well. This might add two additional items to the list above.