Are you both using the same iCloud account? You should have one each. There's no way your purchases should appear on somebody else's account.
But that's by-the-by. If you don't like the basic concept of it, the fine details aren't important.
We might differ on our approach.
In my view:
You want one iCloud account PER device so that way you get 5MB of space for backing up independent of the other devices. When you replace (and I mean sell the old device), then you would use the same account. So
WifeiPhone@gmail.com would be used on her first iPhone, and when she replaces it, she'd login to that same account on the new device, all her stuff gets slurped down and it's like nothing changed, except new shiny case.
You want to SHARE the same iTunes account for making purchases, BUT you need to disable the auto-download feature to all devices because THAT is the the feature that hosed Bullgrit.
The reason you share the iTunes account (and that is seperate from the iCloud account, except for the 1st user), is so that when you buy a $10 app on iPhone 1, it is available to use on iPhone 2 and iPad 1 and iPad 2 in the same household without having to rebuy it 3 more times.
Allegedly, Apple had some other methodology for recognizing the 5 different iTunes users in the house were allowed to share purchases, but I never saw much documentation about it and my way works perfectly (having done it this way since the first iPod Touch and the second iThing entered the house).
I also recommend setting up a SINGLE iTunes Library on a SINGLE computer (unless you know how to manage network shared drive and configure the multiple PCs in the house to share the same SINGLE iTunes library files like I do on my NAS).
Before iCloud, the SINGLE iTunes library was the foundation of how you ensured everybody was licensed to use the same purchases. One accounts buys the stuff (from any device) and one library contains it. You use device profiles and separate play lists (and turn off auto-download) to ensure each device gets only what you want. The device profile part just works, once you disable using the checkbox system for syncing to "whatever is plugged in at the time"
All I know is my way works, and people who don't do it that way seem to end up buying multiple copies of the same music or apps because everybody setup their own iTunes library and account on their own private computers when they got an iThing for Christmas.
On the Hand-wavey Cloud comment: I reckon I'll just have to accept that some folks mistrust things they don't understand. As a technical guy, the "cloud" descriptor doesn't really frighten me because I know there is no such thing as a cloud (except for those white puffy things in the sky). The term cloud was supposed to assure people like Bullgrit and simplify what you didn't really need to understand, and in this case, the exact opposite has been achieved (BG won't use it).
In reality, all Cloud really means is "we got servers and stuff that we own to hold data you own and authorize us to hold for you because you are our customer, and it just works because we spend some time making sure it auto-connects and restores itself in an outage and because it isn't tied up to your single PC that you might throw away.
For example: using Outlook to slurp down email from your ISP (say ATT.net) via POP3 is not-Cloud. it sux. If your PC dies, you have lost a monkey ton of emails that you will never get back.
if you used ATT.net's web interface or instead, used Google's Gmail, you are getting the "cloud effect" in that your email is not stored on a single central, non reliable device. If your PC dies, you just go to your neighbors and use his PC to surf to gmail.com and login and read your email with no interruption. the assumption (that is usually true) is that the guys running the server (AT&T or Google in this example) have beefy hardware with redundancies for power supplies, fans, network cards, etc with back ups running and clustering to make sure this stuff doesn't go down, and if it does, that it can be restored. They are also employing expensive network engineers to manage firewalls and perform security tests that make sure only really good cyber criminals can get your stuff.
This is generally, far superior to what non-technical people are capable of ensuring their stuff is protected on their home PC that they struggle to keep anti-virus up to date, let alone actually back it up off the SINGLE hard drive on it that they've had since they bought the PC.
In more English, the data on your iPhone isn't just "out there" on the internet in the "cloud". Your Apple product is designed to back itself up to Apple's secure, redundant, robust servers. Your Android product is designed to back itself up to Google's secure, redundant, robust servers. there is no such thing as a nebulous cloud that's just out there where everybody's stuff goes. There is a vendor that is backing up that vendor's stuff onto servers housed in that vendor's datacenter that is dedicated to the purpose.
And any vendor getting into large scale storage like that is pretty paranoid about breaches, such that they most certainly dedicate more resources than any normal person can to the protection of it. YOU can steal my home PC and the data off it with far greater ease than anyone can breach a major vendor's network.
Anyway, hopefully I've almost convinced Bullgrit to reconsider.
I do not blame him, or anybody else for being cautious or for mis-configuring iThing equipment in a less optimal way. Even Apple stuff gets complicated, especially once you get more than one user and one device.
With my own non-technical friends, I've got them trained to Ask Me Before They Buy, and See Me Afterwards for Setup. Things go smoother, and I can far more readily avoid some mistakes for them, than deal with them in the "you shouldn't have done it that way" state of affairs.
For that process to work, someone technical like me has to be open to what the friend is trying to do and what they want to buy. I'm not going to just push specific Apple products on them. But I am going to make sure that within the framework of what they wanted to buy their problem is solved, and certain configuration opportunities are done better before they move in and find they goofed up.