Apple watch keeps asking to sign into icloud

I had the same issue, it did have to do with your Apple ID. After a few attempts it asked for the password and once entered it took it and has held up over 4 weeks now.

Same problem

Watch gives prompt to type in on IOS device but iPhone doesn't respond

You were right. Persistence pays off. It was probably 5 or 6 times, but eventually worked. Thanks for the coaching

mine gives me the option of finger-writing my PW on the watch screen or keyboard. The keyboard option then says to use the IOS affiliated with the watch, but no keyboard comes up on screen and if I'm supposed to enter an app to use the keyboard, there are no instructions on where to go. I keep getting a wrong pw notice. : (

From your iPhone>Watch>General>Apple ID>

I've followed this and randomly I'll be asked to login again. What's up with that?

Use your finger to enter your password. It's not that difficult.