No it wouldn't, it allows the user to choose the theme in a setting on their profile page - but there are options: you could set the theme in the database directly using a script and then use css to hide the option to change it from the user profile page.
OR
You could use the user profile fields to add the user's company/institution e.g. in the Optional fields, Institution. You could then detect that $USER variable, add it to the body classes and then use it to adjust any of the css you want to - including adding an institutional logo. You could then just have the single theme and if new companies get involved you can add the adjustments through the customcss setting, rather than creating a new child theme.