It sounds like you're asking the following question:
If I [illegally] downloaded 1TB of songs from Napster in the year 2000, then I pay a 1 time $30 fee to Amazon, does that mean I now legally own all of that content?
IANAL: No, you do not own the content, All these services do is make it so you just have the rights tocapability stream that content, (as if from your computer,), through their service. (Possibly in violation of their TOS.) Once you stop paying that subscription, you can no longer stream that content from them. Your rights to play illegally-acquired content are no different than from before your subscription.