If you're unable to sign in with your Articulate ID in the Articulate 360 desktop app, there could be several reasons for that. Please sign in with your Articulate ID at 360.articulate.com.
Once you're signed in and can see the "Download Apps" button in the top right corner, please check with your administrator if all necessary network endpoints are reachable and if your PC meets all system requirements. After that, you should be able to sign in to the A360 desktop app.
If the button is not visible to you, please contact the administrator of the Articulate subscription. It seems that you haven't been assigned as an active user of a license. The administrator needs to add your Articulate ID under the "Manage Team" tab and the "Team" section (see screenshot). After you confirm this through the invitation email from Articulate, you can sign in to the Articulate 360 desktop app.
An incorrect system time can also be the cause of not being able to sign in to the A360 desktop app. You can find more information about this in this post.