No, PayPal is not FDIC insured. Yes, it works very much like a bank, but as of yet is has not been classified by the FDIC as a bank.
Also, PayPal does not WANT to be classified as a bank due to the restrictions that the fed places on banks (cash reserves, etc.)
Things can change, however. It's entirely possible that the fed can amend the rules to treat businesses like PayPal as banks, or it can leave them alone and keep this strategy of "pseudo" banking separate than traditional banks.
does that mean that Paypal works on an extremely thin fractional reserve?
Not in those terms, but in their 2017 Annual Report, their "Funds payable and amounts due to customers" is about 7 times the amount of cash on hand.
These liabilities are most likely considered unsecured junior debt, so if PP decided to declare bankruptcy, any bondholders would get paid fist, followed by unsecured debt, so it's very possible that the money could be lost.