coinomi supports about 70, exodus seems to support a lot too
http://support.exodus.io/article/96-exodus-wallet-roadmapbut its still only supporting just the top 20 or so coins.
So that must mean that unless all these people are using paper wallets (and i doubt it) these people buying all these alt coins are storing them on online exchanges?? There's no other way right?
There is. You just download the official daemon / core hardware wallet for the designated coin, and store them there.
For example Litecoin,
https://litecoin.org/ Under downloads you can download their official wallet, which is only used for their cryptocurrency.
You can do this for most wallets, use coinmarketcap.com to visit their homepage, and download the wallet from there.
If you are going to download wallets for altcoins that aren't in the top 100 / lower volume / lower marketcap, you should always make sure that their wallet is safe and secure, developers sometimes hide malware in there.
Usually this isn't the case with most established coins, but you never know.