apple tv 2nd generation does not open itunes store

If you mean the legacy model Apple TV (2nd generation) A1378 (2010), then you may need to sign in like this:

For users that have an Apple TV (2nd or 3rd generation) and two-factor authentication for their Apple ID, they may be asked to type the password followed by the six-digit verification code directly into the password field. Additionally, there is a time limit, after which the code becomes invalid.

Try this: First enter the characters of your password (do not submit yet), then manually request a new verification code, then append that code to the password (no spaces or other separating characters), and then submit. That should help you get past the time limit.

I don't have two factor authentication

Some Apple services now require having an Apple Account with two-factor authentication. I don’t have a list of these services, but using Store services seems likely, as money is potentially involved.

Two-factor authentication for Apple Account - Apple Support

Get a verification code and sign in with two-factor authentication - Apple Support

Manage your Apple Account on the web - Apple﹥[Sign In]﹥Sign-In and Security﹥Account Security﹥…

What do you mean it "does not accept your Apple ID"?

What happens when you enter your Apple ID exactly?

Do you get an error? If so, what does it say?

Does your Apple ID have Two-Factor authentication enabled?

I have the same apple tv 2nd gen (2010) and I don't have two factor authentication and since this past week, it's not connecting to itune store either.

I verified everything I can stream music on air play I tried my password 5 times and the verified it on the internet, it's still not connecting.