Sharing Android's Wi-Fi internet connection via... Wi-Fi. Not just extending! Well.. basically what I'm trying to achieve is sharing android's internet connection by Wi-Fi. But Android is connected to internet by.. Wi-Fi. Also, I don't just want to extend the Wi-Fi, I want to share the internet connection because the Wi-Fi I'm connected to has a password you have to input at the Wi-Fi's login page, not in settings as usual. My Android phone is rooted, and I have couple of others if more than one is needed. All of them rooted. Thanks!
You need a phone with 2 sets of Wi-Fi hardware built-in, so that one can be used to connect to your original one and one can share. Unfortunately those are incredibly rare - the only ones I know of are Huawei D2 and Samsung's S7 family.

In your case where your network requires sign-in, consider using your phone as a proxy server and set other devices to use it. Probably works, probably not, but it's the only approach I can think of.