Hahaha, man, it's not Google. They don't give a **** whether you use Chrome or not.
Instead, this has all the hallmarks of someone who has been hacked. You are describing exactly what happens when your account is stolen by someone else. This is potentially a big problem for you if you do any sort of financial work on your computer. You really need to:
- Change all your passwords on every financially-related website you use, or any government site. Immediately. Like, do this today, I'm not joking. Do this from a DIFFERENT computer, preferably one that's ran by someone who understands online security.
- Check your credit cards and bank accounts for fraud. Today.
- Check your credit report ASAP.
- Once all that is done, you should reformat your computer completely and start fresh.
I see that you wrote that you're logged in successfully, which is great, but it doesn't remove the risk that we're talking about here. At the very least you should change your passwords immediately. Good luck!