The default location for the catalog is just a regular folder. You can create an icon to launch LR with the latest catalog. On windows navigate to:
C:\Program Files\Adobe\Adobe Lightroom
Select lightroom (the application) and right-click and choose Send to Desktop.
On Mac OSX launch Finder and go to:
Macintosh HD >> Applications >> Adobe Lightroom >> Adobe Lightroom.app
After launching the app right-click (control-click) on the dock icon and choose:
Options >> Keep in Dock
Files that have logos as their icon have been setup to open a certain program to use the file so that you can double click on the data file to launch the program that displays/edits the file.
LR catalogs are not setup this way by default. It is assumed that you will run LR from the LR program icon and open a new or different catalog from inside the program.
On Windows, to associate LR with the catalog file, you can right-click on the catalog file, select "open", then select Lightroom as the application.
It's probably a Windows thing. I renamed my catalog after it was converted for Lightroom CC. The icon went blank when I did that, and I can't seem to get it to change. It really doesn't hurt anything. But I understand your frustration.
It's a known issue that affects some installs and not others. I have none Mac with a missing logo and other that has the logo. I know other users with no logos on both installs.