I don't think there is a specific term for a company initiative of this sort.
In a general sense, I'd personally include this under the banner of User Research.
In terms of searching for examples of initiatives, you might want to think about terms such as:
- User engagement
- User satisfaction
- User feedback
Most of the efforts in this area typically focus on measuring or improving a particular metric (such as satisfaction). So perhaps search for things like "Examples of improving customer/user satisfaction" - If they display an improvement, the same article may well list the methodology they used.