I believe you should be able to first download Creative Cloud (https://creative.adobe.com/products/creative-cloud) and use this to login to your existing account. Once you do this, you should be able to download Photoshop without buying it again.
I would thin you just do what you did on your first machine. If you getting rid of your first machine you should you should lso sign our from the cloud desktop. On your ne machine all you should need to do is install the cloud desktop application. Open it sign in to your ID with the subscription and install the products you do, Its the sign in the and account with a subscription the does the authenticating and activates products.