I have been working with angular for the last few weeks, and now I have a requirement to dynamically style a public site. The site admin set various color codes as well as a log
It is not possible in that way but rather than using the sass variable, you use the value of the sass variable. It may be any value.
Why? because sass is compiled during packaging and in the end, it would still generate plane CSS.
An example of a framework making use of this optional style processor is angular.
In your case I would recommend looking into dynamic themeing within angular as what you require definitely needs JavaScript. Look into the guide on medium given by one of the contributors.
https://stackoverflow.com/a/54559350/3070499