I was notified of the iOS 10.3.2 update, and when I went to check it out, I saw “Apple Inc.” in the corner. What does this mean? Or can it be displayed by something else?
This means that Apple issued an update for the version of iOS on your device. It indicates that the owner of the software is 'Apple Inc'.
Apple officially changed their name from 'Apple Computer Inc' to 'Apple Inc' in January 2007.
I'm not sure this is the answer you were looking for but it's the best I could do based on how your question is worded. If the answer isn't what you wanted, please edit your question with additional info and I, and others here, will do our best to answer.
The UI has been modelled to be similar to updates provided through the App Store.
For iOS updates, the company is always "Apple Inc.", as they are the only ones who deliver iOS updates to your device. It's shown because the UI is meant to be reflective of the UI when updating App Store apps, in which case the developer may not be Apple.