What do you mean by “paste it in the search tab”? What search tab are you referring to?
The URL is a link that can open a vault or a specific file. But you have to click it in an app that supports those kinds of app URLs.
If you are in an app that supports the URL, then clicking on the URL link will trigger Obsidian to open the note you point to.
I am quite certain Google Docs does not support this kind of link. Strangely, neither does Obsidian as a bare text link. In Obsidian, you would have to format it as a link, like this:
[Your Link Text](obsidian://open?vault=TestVault&file=Test%20Note)