No, unfortunately you cannot stay logged into MAC Check on multiple devices or browser windows. If you log into MAC Check from one device or window, and then attempt to log in from another one without first signing out of the initial device/window, you will get kicked out of the system. If you are planning on using a different device from the last time you logged in, please log yourself out before using the new device.
If you are using a tablet or mobile device, we recommend downloading the MAC Check app in the Google Play Store or the Apple App Store for the best user experience.