Why does bitcoin not have something like nicknames? It would be much easier to exchange bitcoins if everyone could have one or more unique nicknames on the blockchain. Of course what should not be possible is that other people can check the balance by just using a nickname. Wouldn't there be any technical solution to achieve that?
In bitcoin, there are no usernames, just addresses that are derived from cryptographic keypairs. Since an address is derived mathematically, when each node hears about a new transaction, it can simply use the right math equation to independently verify that the rightful owner of an address is attempting to spend the address' funds. This is done via cryptography, using asymmetric keypairs and cryptographic signatures.

In your example, each user could claim a username, but now we have a problem: how will everyone know who owns each username? How can I confirm that you are the rightful owner of your username, and consider your transactions valid? 

A website that has usernames/passwords keeps a central database with all of them stored, so when you sign in, their server can check your password and allow you online. But the bitcoin network does not have a centralized server on which to store usernames/passwords. Instead it is a decentralized network of individual users.

So, you could have every node store all of the usernames, but that could becomes really resource intensive! And also very bad for privacy (remember, all of the blockchain’s transaction are a public ledger). For these reasons (and probably some others), storing usernames requires a more complicated solution for a decentralized network, whereas the addressing scheme used in bitcoin works quite well, even if addresses are tough for humans to read. 

It is worth noting, there have been 3rd party services that allow you to claim a username, and then if someone wants to pay you they can do something like visit a URL (eg. www.nicknamePay.com/sebbo) and be presented with a btc address. In that example, the website could manage you wallet for you, while you can just tell associates to visit an easy-to-remember URL.