My girlfriend and I bought a house last year, we are not married. Can we both claim the property taxes? Both of our names are on the house.

you can each claim half

You pretty much have a choice. One can claim it all or you can split it. It's usually best if only one claims it, allowing the other to use the standard deduction.

But, you have to meet the rules, which are:

1. You are legally obligated to pay it
2. You actually pay it. Paying from a joint account where you made sufficient deposits to cover the payments will usually meet this standard. However. paying from your own account would be a stronger audit defense. 

Since they are not married, they can each only claim what they actually paid themselves (not necessarily half).