is it possible to create an android application that is meant only for internal use? Basically a private application not meant to be installed by non-approved phones?
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Using Google Play Private Channel is probably the best way, but is useful only for Google Apps users.
You can use new Developer dashboard options that let your app be in Alpha or beta phase and is available to only selected google accounts (using groups or google plus circles). Your app can stay in beta indefinitely so not become visible to other users. if you like this method you can also use services like TestFlight.
Mail distribution is also ok it's even mentioned in documentation which I find a little bit odd, but I would advise you to use some file storage like Dropbox and just distribute the link to selected people.