Auto merge of #115436 - GuillaumeGomez:fix-type-based-search, r=notriddle

[rustdoc] Fix type based search

Fixes rust-lang/rust#114522.

The problem was a bit more tricky than I originally thought it would be: we only kept type ID and generics in short, but as soon as there was a full path in the user query, the element didn't get an ID anymore because the ID map didn't know about `x::y` (although it knew about `y`). So for this first problem, I instead always pass the element name to get the ID.

Then a new problem occurred: we actually needed to check if paths matched, otherwise whatever the path, as long as the "end types" match, it's all good. meaning, we needed to add path information, but to do so, we needed it to be added into the search index directly as there was no mapping between `"p"` and `"q"`.

I hope this explanation makes sense to someone else than me. ^^'

r? `@notriddle`

Author: bors