Browsers open all links in a new tab
To my thinking, a logical place to start and fix this is focus on Safari:


Go to the settings and change the tab behavior, click a link, then change it back to your preference if needed. 
Next be sure the issue happens after you log out and log in again. 
Next check is you have custom extensions or CSS or malware that overrides the css for all pages - this setting can be hinted from the web page, but should respect your settings in browser. https://stackoverflow.com/questions/15551779/open-link-in-new-tab-or-window


Some oddball causes:


Faulty keyboard with a modifier key held down. Could be the main keyboard or any other usb keyboard, screen sharing software or Bluetooth keyboard. 
You’ll need Apple’s keyboard viewer or other tool to visualize which modifier is held down of simple trial and error doesn’t help.
Make a new user to see if the apps work with clean user default settings and preferences. 
Reinstall the os if you have no luck with anything else. Consider making a backup and doing an erase install - test browser before you load your apps and settings.