Many websites use various means to identify specific users. The IP (proxy), browser fingerprint, cookies and so forth.
So let's say you're managing 5 Instagram accounts from your home computer. All of the accounts will share the same IP, fingerprint, cookies etc. Now so long as you stay under the radar and don't do anything to get banned, you're fine. However, all of your accounts are now linked... so if you put 1 account at risk, you potentially put all of your linked accounts at risk.
This is why you should not only manage accounts with proxies, but make sure they're created with dedicated unique proxies. Proxies alone are not enough either. Let's say you use dedicated private proxies, but login to each account via the same browser on the same computer. There's more than enough info leaked to identify you as one person (fingerprint, footprint, browser WebRTC IP leak). This is why people also use software to manage multiple accounts along with proxies, as software made by a competent developer should keep everything self-contained on a profile by profile basis. This is absolutely essential if you're managing multiple accounts (whether it be 5 or 5000). If you're building up 5 accounts and breaking TOS on one of them... that could get you banned and put the other 4 at risk too.
Datacenter and Residential proxies: This is also something to consider. You can buy datacenter proxies from most providers for $1-2 each. So long as those IP ranges have not already been blacklisted by the websites you wish to use them on, they're fine for many uses. However, an increasing number of websites (including social media sites) have become aware of the use of datacenter proxies. They're much easier to spot as proxies and I suspect (although I haven't researched this), many websites will assign a trust score to your account based on the IP used, how you signed up to the account, inconsistencies with the browser.... outdated version, proxy geolocation time doesn't match with system time, suspicious behaviour and so forth. I have no evidence to support this though, so don't take this bit as fact. I manage a number of social media accounts with datacenter proxies and so long as they're dedicated (not semi-dedicated i.e. shared with 3+ other users that might also be using them on the same website), I've mostly been fine. However, I get sigificantly less blocks on websites such as Instagram when using a homemade residential proxy using a data-sim card + tethering. I need to find a cost-effective way to scale this up to create multiple residential proxies without buying 20 phones or dongles - so if anybody has a solution, I'd be most appreciative!