You do. Ulys is non-custodial: your funds stay in your own wallet, and Ulys cannot move them without your permission.
Your wallet uses Privy, a Stripe company. Privy splits your key into encrypted pieces and keeps them in secure hardware, so you do not need a seed phrase. You log in with Apple, Google, email or your phone number.
When you turn on Earn or create a smart order, you give a limited permission for that action only. You can remove it in the app.
If you would rather hold your own copy of the key, you can export it. That is optional.