As soon as I opened the game, it took me to an external tab where they ask me to enter my Patreon email and password. Is this normal? Is it a bug?

Yes, that's normal. The game is free-to-play, but for a Patreon subscription (even a free one), the game gives bonuses. You can log into your Patreon account in advance. You just need to grant access to the app so the game knows your subscription level. The higher the subscription, the better the reward. It's not the game that asks for login and password, but the Patreon website.