Google-chrome is famous for its problems with .desktop files and their icons, especially if you had an other version installed before.
Since you installed it from a .deb file, an icon must be installed. Most certainly, there must be an old, outdated or incorrect .desktop file in ~/.local/share/applications causing the "disconnection".
Local .desktop files "overrule" global ones, therefore, if there is a local .desktop that does not "match" (the current version of) a globally installed application, a variety of symptoms can occur. An iconless appearance is one of them.
What do do
Since you technically don't need local .desktop files of a globally installed applications (unless you want do make modifications to its behaviour), you can safely remove all .desktop files, referring to google-chrome, in ~/.local/share/applications. Simply browse to the directory and remove all .desktop files of google-chrome or webapps, calling google-chrome. Then log out and back in and run the application again to see if it is fixed.
If you are having doubts on a specific .desktop file, make it executable to make it show its icon and the correct "interface" name (as defined in the line Name= inside the .desktop file).