Hello @Matt and welcome to our community!
If the asylum process of your girlfriend ends with a positive result she will get a passport that allows her to travel. She can come to Greece with it in order to get married to you. Probably there are rules that enable her to get a residence permit for Greece through this marriage - at least we got a law like that in Germany. Maybe you can do some research about that on your own.
In order to get married you usually need a number of documents. At least your girlfriend got a passport, but for example in Germany you need a certificate of birth or a certificate of no impediment amongst others. This might be the biggest obstacle in this case. It’s probably best to talk to registry offices in Greece to find out what you need to do to get married - they presumably know best.
Marriage in Spain seems to involve a lot of bureaucracy. And if both of the spouses are not Spanish citizens, at least one them must have lived there legally for a minimum of 2 years. I don’t know how long your girlfriend has been in Spain now and if the time she spend there counts as “legal residence” because she is an asylum seeker without a residence permit. Anyways, there is also a couple of other things to keep in mind. Check out the following link for further information: