No, it is possible to store several different cryptocurrencies within certain wallets. This means that you won't have to get a wallet for each altcoin, but can instead stick with a few ones. These multiwallets typically allow you to store a certain selected coins and tokens within the wallet, all while controlling the private keys yourself, making it very secure.
Some coins (such as STEEM) does not really work well with multiwallets, so you will never be able to find STEEM in one of these. I'm not an expert on lesser used altcoins, but I'm sure there are a few that won't fit into these multiwallets, so you might have to download a unique wallet for some of them.
Anyway, there are two different multiwallets that I recommend users:
1) Coinomi. This has some of the big ones like Bitcoin, BTC, Dash, in addition to a huge amount of tokens and smaller coins. It brags to hold over 500 different crypto assets, so you might fit a good chunk of your portfolio into this. Download it from https://www.coinomi.com/.
2) Exodus. I like Exodus better, since it holds most of the major coins and tokens, including EOS, Ethereum (+ Classic), Bitcoin, Litecoin. etc, and it also has some of my personal favorites. It "only" supports something like 90 coins or tokens, but I find it to have most of what I am interested in anyway. Download it from https://www.exodus.io/.
Above is a screenshot from the Exodus wallet, showing what it looks like.
Another different solution to not having to download a lot of wallets would be to store your crypto on an exchange. This is very easy, but it has some very big potential problems. For example, what happens with airdrops? What happens if they lock your account, or the entire exchange gets hacked? I would advise against it, but I would also say that it's fine if it's only for holding a very small amount of coins that you would not be too upset about losing.
Depends on how many you have. Some wallets like the nano S will hold 6 to 12 coins. You can also leave them on the exchanges but you are at greater risk of loosing them.