I'm not quite sure what you mean by the new tab interface, but I think you want to go into settings, and under the Search section, look for "Search Suggestions" and then toggle "Show search suggestions ahead of browsing history in address bar results...
It doesn't completely solve the issue, but you can right-click and select "show in folder".
This is a duplicate of https://connect.mozilla.org/t5/ideas/searching-for-a-bookmark-show-the-folder-where-the-existing/idi-p/8655
Generally the website controls where links are opened by default. It may be that the search engine is choosing different options in different browsers.
If a website isn't opening links in a new tab by default, then you can press cmd (on Mac) or ctrl ...
