Profile not signing in on Xbox 360 games on Xbox one

Signing in on Xbox 360 would actually not affect your Xbox One sign-in. You can use the same profile on Xbox One and Xbox 360 at the same time without conflict.

It does sound like your profile is being signed in elsewhere and that's forcing a sign-out, so a password change would be a good precautionary measure.

On Xbox 360, you can have your profile stored on a flash drive and sign in that way. On both Xbox 360 and Xbox One, you can require a pass code to sign into a profile as well. Essentially, they would have to sign in with their MSA and password, and then
enter the Pass Code to verify they are the owner of the profile.

I signed in to both the xbox one and the xbox 360 game. Instead I get an error saying I need to download the profile.

because all 360 games use the old 360 systems, you need to create an app password in your microsoft account so that you can then sign in to the 360 subsystem.

How to set up an app password to sign in to your Xbox 360 console | Xbox Support

you need this to sign into a 360 console or sign into a 360 profile to use 360 games on xbox one or series consoles.

To play games, even on Xbox 360, it has been required that you're signed into an Xbox Live profile (aka Microsoft Account). It's possible that previously it was configured to automatically sign in, and you didn't realize. It's possible that the account
that had auto-signed in removed the device from their account or changed security settings, and would require reauthorization to sign in.

Now, that is speculation on what might have happened as to why you're seeing a sign in screen unexpectedly. But the point would be that previously when using the 360, there was an account/profile being used.

This page here has details on how to add a profile to your Xbox 360. That would be for if there isn't
one present. Do you see if there is a profile already on there?