The stay signed-in feature is currently available only on the mobile app if you use multi-factor authentication during sign-in, where you sign in with either a passkey or a password with an SMS OTP. If you only use a password to sign in, you will need to sign in again next time.
If you have not registered a passkey yet, visit ‘Account Overview’ > ‘Edit Profile’ > ‘Account Security’ to create a passkey for your account.