I went to eos.io looking for a download link to download the EOS blockchain and create a wallet like how Bitcoin Core works. Or does EOS doesn't work like this as its a POS system and there is no client software?
The wallet tool provided in the EOSIO software is called
keosd and can be downloaded from GitHub or Docker registry together with
nodeos (node software) and
cleos (cli software to interact with
Image from here
To use the tools from docker, you can follow the article in the docs. Alternatively, you can install the tools locally by cloning the GitHub repo and building the toolkit as explained in this document.
Once you have
keosd, you can use the wallet article in the docs for example usage.
For a more comprehensive walk-through, the official docs detail a step-by-step process here.