iPhone 13 is connected to printer via same WiFi network but phone doesn’t recognize printer

Hi,

Is the printer an AirPrint model (see the list in About AirPrint - Apple Support)?

If so, in case of difficulties with AirPrint, and if the setup is OK, a problem with the router not correctly discovering/registering/detecting one or more devices can be suspected. You may want to try the following procedure (in the exact order):

Power off the Wi-Fi router, the printer, the iPhone, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the iPhone. Finally, power on all other devices/computers.

Use AirPrint to print from your iPhone, iPad, or iPod touch - Apple Support