That's a tough one.
Technically speaking; StyleMart and CleanMart are apples and oranges. They are completely different things.
With "StyleMart" we have the freedom of using modules and other JS&CSS without being afraid of possible conflict caused by the 3rd party extensions/templates. Because StyleMart is a "all-in-one" package. People usually don't install anything on top of StyleMart.
But CleanMart is a VM Template only, and it gets integrated to an another Joomla Template. There are many more extensions/plugins/templates will be installed on top of CleanMart. The "Login Popup" you like on StyleMart comes as a login-module with StyleMart (but it is integrated to StyleMart and can't be used as a standalone). So, it can't be installed to Joomla+VM that has the CleanMart.
But I will recommend you to install the CleanMart on top of your Joomla Template and find a standalone login-module for Joomla (it is not supposed to be for VM specifically, it can be for joomla) that looks & works similar to what you see on StyleMart template.
Take a look at these:
extensions.joomla.org/profile/extension/access-a-security/site-access/poplogin
extensions.joomla.org/extension/bt-login
use google.com to find more.
Regards.