Can no longer access my Azure subscription after client adds me as admin to their account
There are two issues here.
First, to access your old subs you need to switch the tenant you are looking at in the portal, currently you are looking at the clients tenant, which does not have your subs. Click on the "directory and subscriptions" menu in the top right and select your tenant. You should then see your subs.
The second issues is with access to your clients sub. The client needs to create your user as a guest user in their AAD tenant as documented here.