As of now, Google has not provided any method to hide this option. However, there is a workaround that can be used to Disable this feature. Hope it will help you.
Please note that the user switching cannot be hidden, only way is to disable it. The trick is available from within Chrome itself. Follow the next steps-
Type chrome://flags in the address bar. Find Enable New Profile Management System.This option is set to Default by Chrome. Select "Disable" from drop down list.Restart Chrome, and if needed reboot your PC. ( not necessarily). You are welcomed with the older options back. If, however, you need the switching enabled, you can get it back by following same steps.
Please note once again that there is no way to hide the profile menu, only way is to disable it. Refer to this Google forum where the person from the team has confirmed to it.