Why does my custom launcher open under a separate icon?

If an application produces a second icon, not being a generic one

If an application generates a second icon in the Unity launcher other than a a grey, generic icon, there must be already an (other) existing launcher, calling p4v in its main command:

An application's .desktop file by definition has a "main" command in the first line, starting with Exec=. This is the command that is run when the icon is clicked in the (Unity) launcher. Apart from that, a .desktop file can have one or more shortcut- sections, available when you right-click on the representing icon in the Unity launcher.

Multiple .desktop files calling the same application in their main command

The symptoms you are experiencing can happen when more than one .desktop file in either /usr/share/applications or ~/.local/share/applications have the same ("main-") command in their first Exec= line, but the files are differently named.

P4v by default should install (and probably has installed) a .desktop file, named p4vasp.desktop in /usr/share/applications.

How to fix
Your problem should be fixed if you remove your own (probably local) .desktop file, then run p4v from Dash and pin it to the launcher.

What if you want to customize the existing launcher?

If you want to change the .desktop file's behaviour or e.g. its icon:

• copy the file from /usr/share/applications to ~/.local/share/applications
• don't change the file's name
• open the file in gedit (drag it over an opened gedit window) and make the changes you'd like
• log out and back in, the local version of the .desktop file will overrule the global one.

To revert, simply remove the local one, log out and back in.

Related to this post

Solution that worked for me on ubuntu 18.04 when struggling with getting an AppImage added to my favorites. This solution is summarised from https://beamtic.com/duplicated-icons-in-launcher

1. Get window class of your program with double icon by executing

   xprop WM_CLASS
   

2. Select the main window of that program by clicking on it. I received two strings and just took the first one (lower case).

3. Add the following line to the .desktop file (I had my .desktop file in ~/.local/share/applications) and replace [Name_of_Window_Class] with the output of the xprop command:

   StartupWMClass=[Name_of_Window_Class]
   

4. Start/Restart the application, done!