What name did you expect it to be? On Nougat and earlier the Bromite webview will show as Android WebView. In Oreo or Pie and higher, if Bromite webview is installed then the Bromite name will show.
Since the /e/OS webview is a fork then it won’t show the Bromite name obviously. Use a webview test app or a webview-based browser to test the useragent.
If it shows your device as something else then you know the Bromite fork is used. My Essential PH1 shoes up as a Samsung device for instance.
Since the webview is just an engine (based on Chromium) it doesn’t access anything as far as I know. Browsers and apps that use it are the ones sending data through the engine.
Standard Android webview, Bromite Webview, and Ungoogled System Webview can be trusted to not do anything on their own. IMO. Can’t vouch for Google’s webview.
There is also TrackerControl, but I think Blokada provides more info. They both (I think) come with a default blocklist, but you might want to add a couple more lists from the menu. (Exodus list is in there, too.)
Alternately, setting up a Pi-hole instance on your home network (on a Raspberry Pi, for instance) can reveal a lot of background activity from wifi-connected devices.