Does anyone know what this grey app icon is?

When passing your pointer over that anonymous Dock icon, does it show a name above it? Did you run any script to add an item to your Dock?

Drag and drop this AppleScript into Apple's Script Editor, click the hammer icon, and then click run. It will pop a dialog with a list of all application names in the Dock in the order of their appearance in the Dock from left to right. It excludes Finder.

# return a list of application names in the Dock in left to right order
# reference: https://stackoverflow.com/questions/14245301/can-applescript-list-all-the-applications-placed-in-the-dock
set plistpath to (path to preferences folder as text) & "com.apple.dock.plist"

tell application "System Events"
	set plistContents to contents of property list file plistpath
	set pListItems to value of plistContents
end tell
set persistentAppsList to |persistent-apps| of pListItems

set dockAppsList to {}
repeat with thisRecord in persistentAppsList
	set end of dockAppsList to |file-label| of |tile-data| of thisRecord
end repeat
set {TID, AppleScript's text item delimiters} to {AppleScript's text item delimiters, return}
display dialog (items of dockAppsList) as text with title "Apps in the Dock"
set AppleScript's text item delimiters to TID
return dockAppsList

Unlikely.

Check System Settings > General > Login items for anything suspicious.

You might also want to disable Settings Settings > Desktop & Dock > Show recent applications in Dock.

Maybe someone can figure out what it is with an EtreCheck report.

Install EtreCheck and post the report. Allow Full Disk Access.

https://etrecheck.com/

Use the Additional Text icon to paste in the report.

What about under "Allow in the Background"?

Posting a shot of the icon. It's rather non-descript.

Since I don't know what it is I can't answer that.