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I've had this problem since Lolipop upgrade, my wife is now getting this problem after moving from Z1 compact (lolipop) to Z3 compact too, also seen this issue on XDA forums so it's a definite Z3 issue.
If I click a link in an app (Twitter, Google Search etc) then a Chrome tab opens but nothing loads in it, clicking the address and pressing return does nothing, copying the address and pasting in to a new tab or incognito window works OK. killing the app in the mutitask window doesn't help, restarting the phone allows it to work for a couple of days before it starts doing this again.
A solution is needed and rebooting the phone is not a solution, it's a temporary fix. I have a few Gb free on the phone, all other apps working fine. I'm on my 3rd Z3 in a few months, it's done it on all of them, resetting the phone to factory settings does nothing after a few days.
Do you both have the same Chrome app version?
Do you have any antivirus or taskillers?
If it works on incognito mode, then there might be an issue with your google account if you are syncing it with your desktop
Check the Chrome version on all of your devices and post it here, the latest update for one device might not be the same on another device, also follow phy's advice and uninstall the updates and don't reinstall them for few hours.
Keep an eye on that and request them to post their Chrome version, it is not a Sony/Z3 issue, Sony has nothing to do with Chrome, this compatibility issues and or bugs have to be addressed by Google, even if an app, game, etc works with a phone, it doesn't mean it will work with others, different phones and as you can see, different Chrome versions. that simple.