I was asked this question recently, and after a little thought gave my answer as “both”. There are, of course, other views.
To make a difference, or be in any way meaningful, agile has to have some tangible impact on the real world. If it doesn’t do that, then it’s only ideas. We can debate how positive or negative that impact is, but there are real practices that influence how works gets done. We might also debate whether “methodology” is precisely the right term (or is it a way of working, or a general approach, or…?) but in terms of words on offer here, “methodology” is good enough. Most of all, it represents something real.
I dislike the idea of something being a mindset for the reasons above—anything’s got to have an impact on the material world to make a difference, and I don’t like aligning myself with something that might be so “immaterial”. On the other hand I have to admit that working for so many years in this way has influenced my thinking in many areas. For example, I always try to focus on the value or purpose of something beyond the mere action itself. I also try to break big ventures into much smaller tangible tasks that provide meaningful progress. Both of these are very “agile” things. So it does provide a particular world view, at least for some, and therefore could reasonably be described as a mindset.
Does it have to be both? It seems acceptable for it to be a methodology for some without being a mindset. And again, the mindset has to be turned into actions to be meaningful. But for me at least, I don’t feel the need to choose between the two.