Describe the workflow you want to enable

Hello! I'm trying to use metadata routing with BaggingClassifier and BaggingRegressor however it is implemented for the fit method, not the predict one. I am wondering if there is a particular reason for not doing it on the predict function or if this is a feature that could be added. This would enable situations like the following, which currently gives an error:

import numpy as np
import sklearn
from sklearn import ensemble
from sklearn.base import BaseEstimator

sklearn.set_config(enable_metadata_routing=True)


class CustomEstimator(BaseEstimator):
    def fit(self, X, y, foo):
        return self

    def predict(self, X, bar):
        return np.zeros(X.shape[0])


estimator = CustomEstimator()
estimator.set_fit_request(foo=True)
estimator.set_predict_request(bar=True)
model = ensemble.BaggingRegressor(estimator)

n, p = 10, 2
rng = np.random.default_rng(0)
x = rng.random((n, p))
y = rng.integers(0, 2, n)

model.fit(x, y, foo=True)
model.predict(x, bar=True). # TypeError: BaggingRegressor.predict() got an unexpected keyword argument 'bar'

Describe your proposed solution

Similar to the fit method, something like:

if _routing_enabled():
    routed_params = process_routing(self, "predict", **predict_params)

However, I don't have enough understanding of the metadata routing implementation to know exactly what should be done.

Describe alternatives you've considered, if relevant

No response

Additional context

I tried to have a look at the history of PRs/Issues to find a discussion around this point, but could not find it in the PR introducing the metadata routing to these estimators (#28432).