Bitcoin is not a payment solution.
It's more a currency or a commodity or a virtual good.
The dollar and the euro are not payment solutions either.
It's both. The Bitcoin client software and p2p network is a payment system. The unit of account (bitcoin) that works on that network is a currency.
Sometimes I wonder if it would've been a better marketing/branding decision to use two names for these components - something like "Bitnet" for the p2p network and client, and "Bitcoin" for the currency. Not sure...