Chat/ Omni-Channel is disconnected when the Console browser tab is refreshed

This is expected behavior. Refreshing the entire Service Console's browser tab can disrupt the Chat/ Omni-Channel session and terminate it, even though the Agent did not toggle Offline. The refreshing of the page can interrupt the connection between your network and the Chat servers, thus making it appear that the Agent is no longer online and causing a connection error. 
Please only refresh the frame of the page that you are targeting, such as a record or a list view, by clicking on the appropriate Console refresh buttons. 

As of the Summer '19 release, a new feature was added to the Omni-Channel settings to help agents avoid accidentally ending conversations in progress or unintentional logouts.
To enable this feature, go to the Omni-Channel Settings in Set Up and check the box "Display a login confirmation upon loading a console with Omni-Channel".

Please refer to the Release Note 'Prompt Before Logging Out Agents of Other Browser Tabs' for additional details.