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 (on Windows/Linux) whilst clicking the link to have it open in a new tab.
Your preference typically is saved in a cookie, so if you set Firefox to clear cookies when it closes, you'll need to return to those pages each session. Otherwise, your setting should persist.