I'm really tired of all these nonsense fights on whether Agile is a methodology or not. Why would anyone care? It's just a tool to manage th project, in a slightly different way from Waterfall (iterative/cyclical).
If you ask any Agilist, he will say that Agile is indeed a project management methodology, if you ask any Watefallist, he will tell you that Agile is just framework. If you ask a real project manager (who is mature in his profession) he will tell you this: "I don't care".
Anyway, I will answer your question, and to do this, we will need to understand what a methodology is.
A methodology defines the project lifecycle and a standard project management process that needs to be followed by the project manager, stakeholders, and team members in order to finish the project.
Does Agile have its own lifecycle? Some say it does, some say it doesn't, but one thing is for sure, there is no standard Agile lifecycle (If I'm mistaken please point me to a life cycle that everyone is following).
Dos Agile have its own project management? Sort of, but again, there are is no standard.
Does the definition of a PM methodology above apply to Agile? Obviously not. So Agile is not a methodology, that's why it's better to say "Agile Software Development" and not "Agile Project Management", because Agile falls under regular (Waterfall) Project Management, with some tweaks.
PS: I'm sure that...
- Agilists reading my answer will oppose it.
- Watefallists reading my answer will agree with me.
- Real project managers won't care.