Open new links as child tab

Not sure i understand the question...
Note that you can open links in new windows and tabs without restoring them in the TO using the Ctrl & Shift clicks on url nodes in TO tree

eg, from a google search tab, you open some links in a new tab.

These tabs should be children of the google search tab.

That's how it works in firefox.
So it creates a self-organising hierachy.

Do you have the Tree Style Tabs option enabled in TO options? As when it's enabled it is actually work exactly as you describe.
This option is enabled by default for all new users. But if you use TO for a very long time (something like more then a year) it might be disabled, in your case.