If I am on chrome and am looking up a product on Amazon on tab A, can they tell what I search if I open a tab B while tab A is still open? Should I use a new chrome or another browser window to increase security? My thought is that they can only tell if you switch away. Right?
Amazon can confidently tell what you you're looking at on another tab if the site you are looking at is affiliated with or is Amazon themselves.
Amazon can also tell if you leave focus of your tab to view something else - see this stackoverflow answer.
They cannot tell the contents of or what you are searching on another tab if that site is not affiliated with / sending that information to amazon voluntarily.
You can prevent amazon from being able to see what you're searching on an Amazon website or website that is actively sending information to Amazon in tab B by using a separate browser with a different fingerprint, cookies, and IP address (through a proxy or VPN).