I purchased an app on my iPhone and the app is also on Mac
Is your question about an iOS app which has an equivalent macOS app? 

In this case, they are really two separate apps with the same name. 

Purchasing or getting the iOS app on your iPhone does not automatically give you the macOS version on your Mac.

You must go to the macOS App Store and download the macOS version.

Examples of apps I use that meet this criteria:

Evernote, Google Chrome, Firefox, Google Earth, Dropbox, Kindle ... and many, many more.