Yes, it is subsidized. The reason is that being able to send and receive mail cheaply and reliably is the cornerstone of our society and our economic system.
When people talk about International Mail being "subsidized" they're talking about EG treaty negotiations with China that force the USPS to receive international packages from China at significantly below cost. They then have to pass those costs onto the US consumer.
I've not seen any citation which claims the junk mail part of the USPS is unprofitable.
To be clear, I hate junk mail. I've just seen this claim made a few times without citation.
The USPS was founded in 1792 and became self-funded under the Postal Reorganization Act of 1970 almost 200 years later, so, no, it hasn't “always” been required to pay for itself.