I ask questions cause I wanna know!
Seriously, I suspect the answer to this question is as varied as there are people who'll respond to it.
To me, spirituality in budo is human nature and observing it in ourselves and others which helps us to have more info with which to make decisions. Does this include religious aspects? For me it does, but I think it depends on what religion someone follows, if any. It has a lot to do with how we treat each other. Hopefully as we learn we can develop compassion and treat each other with respect in ethical and equitable ways both in and out of the dojo.
Budo training is like a magnifying glass which allows us to look at ourselves and others in ways that are often hidden. At times this can be painful and reveal things we would rather not face. How we deal with these depends largely on qualities like courage and determination, plus a strong support group. Our family, teacher, dojo training mates, etc. make up the paradox of our practice which involves just us and our innards but can't be done without the others participation.
I'll let some others (hopefully) add some stuff...