Yes it's entirely possible for the WIFI router owner to see what websites you're visiting and see your IP address of your computer or device connected to the WIFI router.
If a WIFI network is left open and it's a neighbors WIFI connection then they'll likely never bother or may not even know how to look up the router logs.
But for businesses that have WIFI they may look at the router logs from time to time.
If it's a friend or relatives house they can also see the websites you're visiting if they wanted too.