Can I use one Apple ID for Apple Music and another for iTunes & App Store?
UPDATE for follow-up question from OP

You can Authorize each other's computers on your Apple Account. There is a limit to doing this as you can only authorize 5 accounts on any one Apple ID.

Then in iTunes on you computers, you can log into one account, download all of the apps associated with that account. 

Then log out, and log into the other Apple ID and repeat the download. This gets around the shared drive, but doubles the amount of data you need to store.

When an app updates, you need to be logged into the account that "purchased" it for DRM reasons to update. If it is one of the apps on you account, then you can update on your iPhone through the App Store. Your friend will have to log into your Apple ID an update in iTunes, then sync their iPhone to iTunes.

When an app updates on your friend's account, they can update however they want to, but you will need to go into iTunes, log on to their Apple ID and update that way, then sync your iPhone with iTunes. 

As iPhone is pretty boxed in, I don't think it takes to kindly to switching back and forth between Apple ID's. I am not even sure if you can do that without it resetting the iPhone, and I am almost certain there isn't an option to have two Apple ID's logged into one iOS device at the same time.

So once you have done all that, you have probably violated 42 conditions in the TOS. Just don't let Taylor Swift find out.



Prior Answer:

The short answer is yes, but it is a bit of a hassle. 

Each person can authorize the other's devices on your Apple ID, though there is a limit of five devices that can be authorized. So if you have computer, iPhone, iPad, IPod, then there isn't much room for your friend's devices. If you live in the same location, you can put the media on a shared drive and bring it into each others iTunes catalog. 

Here, the only catch is that the apps will only update for the person who bought the app, so you either need to have their Apple ID and password to log in as them to update on your devices, or with some maintenance on a regular basis, you can manually delete from your catalog and re add the updated app on your iTunes. I did this with my girlfriend long before Apple set up Family Sharing, which leads to...

How much do you trust this friend? You could set up a Family Sharing under iCloud that allows you to share everything, including media from iTunes and the App Store, plus photos, calendars, etc.

The catch is that the person who sets up the account is responsible for all of the purchases on that account. Only one credit card can be set up per family share, so basically if you set it up you pay. It is designed for a head of household situation with a spouse and three kids, so it is a little dicey if you don't trust your friend financially.