MNT Deprecate metrics.pairwise.paired_*_distances and paired_distances public functions

[Shreesha3112]

Reference Issues/PRs

Fixes #26982

What does this implement/fix? Explain your changes.

Deprecates metrics.pairwise.paired_*_distance functions and metrics.pairwise.paired_distance

• [x] metrics.pairwise.paired_euclidean_distances
• [x] metrics.pairwise.paired_manhattan_distances
• [x] metrics.pairwise.paired_cosine_distances
• [x] metrics.pairwise.paired_distances

Any other comments?

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[Shreesha3112]

I've noticed that the changes to metrics.pairwise.paired_distance could impact cluster.AgglomerativeClustering. Specifically, the linkage_tree function relies on metrics.pairwise.paired_distances when the affinity isn't precomputed. Given this, should I go ahead and make the required changes to both metrics.pairwise.paired_distances and the linkage_tree function within this PR, or would it be preferred to handle them separately?

[ogrisel]

Maybe we could put the code in private functions (with a _ prefix and without redundant input parameter and data checks) and make the deprecated public functions call their private counterparts.

The AgglomerativeClustering estimator could then directly call the private functions.

[glemaitre]

I resolved the conflicts with main. I will give a round of review.