Definitely, Bitcoin can be one of the biggest rival of PayPal and this is simply, because Bitcoin is way much better than PayPal, like in PayPal there so many barriers while transferring money from one person to another, it can never give you the amount of privacy one enjoys in Bitcoin.
There really aren't that many barriers. But indeed, privacy is superior in Bitcoin. There are altcoins which provide even better privacy, though. It's very important to be careful about UTXO selection with Bitcoin if you want to maintain anonymity.
Also bitcoin being decentralized, nearly available in every country and having lower transfer fees. So much plus points one gets when transferring money from Bitcoins, so PayPal feeling the pressure is common as they simply cannot match the level of Bitcoin.
In many cases, there are no fees to use Paypal. For instance, US users can send money back and forth (including to and from bank accounts) for free. Again, I don't think that Bitcoin and Paypal are targeting the same markets. For the foreseeable future, Bitcoin is not viable for high volume remittance. And even if it was, Paypal was never particularly competitive in this space, hence the OP.