What problem does this address?

Sometimes, two pages can share the same Page Title because they are same page names located in two different parent page.
Example of page structure:

Team1
       Our references
Team2
       Our references

Depending on the website architecture, this can be relevant.
The issue is that the Link Inserter doesn't make edit difference between "Our references" links, so people will need to try them to see if it's the good one.

What is your proposed solution?

For complex needs, it would be nice to be able to override the LinkControl results and show the name of the direct parent page as the inputLabel of the link search results.

At least, a way to hook this without overriding the whole component would be fine, but I'm also wondering whether adding the parent page would be also nice in general as this is a pretty useful information when searching a page.