Status Icons

From your description that's Airplay. See this Apple support article-->How to AirPlay video and mirror your device's screen - Apple Support

That is a “share” button. Tap on it and you will have several options for what to do with the page you are on.

The share icon is not in the status bar, and it has an arrow going up from the center of a square.

Thanks, now I see I was looking in the wrong spot.

I agree with Lawrence. Its the Airplay icon.

Sharing icon does not appear in the status bar next to the battery for any reason that I know of.