Stable extender contract via `fit` / `_fit` resp `predict` / `_predict` separation

### Describe the workflow you want to enable

tl;dr, I am suggestion to refactor `scikit-learn` internals to a layer separation with boilerplate between `fit` and `_fit` resp `predict` and `_predict` methods, to make extender interfaces more stable. Also see https://github.com/scikit-learn/scikit-learn/issues/31728

More background: Currently, every time `scikit-learn` releases a new minor version - e.g., 1.5.0, 1.6.0, 1.7.0 - compliant extensions, e.g., custom transformers, classifiers, etc, break - specifically, referring to the API conformance as tested through `check_estimator` or `parametrize_with_checks`.

These repeated breakages in the "extender contract" contrast the stability of the usage contract, which is stable and professionally managed.

For a package like `scikit-learn` which means to be a standard not just for ML algorithms but also an API standard that everyone uses, this is not a good state to be in - "do not break user code" is the maxim that gets broken for power users writing extensions.

Of course maintaining downwards compatibility is not always possible, but nothing should break without a proper warning.

### Describe your proposed solution

The `fit`/`_fit` separation would ensure stability of the extension contract - and would also allow to build secondary deprecation patterns in relation to it.

The (oop) pattern this would implement is the so-called "template pattern".

It would allow to remove likely changing parts such as the boilerplate (e.g., `validate_data` vs `_validate_data` and such) from the extension locus, and thus completely prevent breakage in relation to boilerplate changes.
Reference: https://refactoring.guru/design-patterns/template-method

Examples of how this can be used to improve stability:

* `sktime`, for a different API, has a separation between `fit` calling an internal `_fit`, where change-prone boilerplate is sandwiched between a stable user contract (`fit`) and a stable extender contract (`_fit`); similarly `predict` and `_predict`
* `feature-engine` overrides the `BaseTransformer` `scikit-learn` extension contract with a similar pattern using `super()` calls in `fit` etc.

The proposed `fit`/`_fit` pairing that combines strategy and template pattern can be introduced easily via pure internal refactoring - this also add a layer of tests that checks the stability of the extension contract.

As a side effect, this would likely reduce the amount of copy-pasted boilerplate among the core estimators in `scikit-learn` proper as well.

### Describe alternatives you've considered, if relevant

The obvious alternative is "do nothing" - which will lead to packages with extensions producing their own version of mitigating patterns for the unstable status quo.

The problem with this is that the extending packages are mutually incompatible with their extender pattern. It would be better if there is a single stable extension pattern prescribed - or at least recommended - by `scikit-learn`.

### Additional context

_No response_

COMMENT:
A few points on this issue.

Regardless of whether we implement separation of `fit` vs `__sklearn_fit__`, I'm not sure how much that'd help with stability of the API. We have quite a stable API when it comes to what we've marked as developer API, which is loose at the moment and needs work improving. I can't think of a previous case where this would have helped, except usage of `validate_data`/`_validate_data`/`check_array`/etc.

As for what could go in `fit`, we have some options such as:
- constructor arg validation, which are inside a decorator right now;
- input validation, based on an extended version of tags where validation parameters are defined there;
- metadata validation / routing, both in consumers (which doesn't exist at the moment), and meta estimators, which includes some boilerplate code at the moment.

However, this seems like something where we might need a SLEP, and a maintainer needs to champion that. However, it is possible that we can find a backward compatible way to implement the change which would avoid going through a SLEP.

That all means this probably would be a part of our longer running project "Developer API", and if there's consensus, we can include it in the next release of the one after.

cc @scikit-learn/core-devs for visibility since this would touch every estimator.

Labels: RFC, Developer API