I love questions like this.
I take it that by "spiritual" in the context of budo trining, people don't mean "conducive to the promotion of the immortality of the soul." Rather, they mean something like "conducive to a feeling of emotional and psychological fulfillment, endowing life with significance." Suppose two people spend an hour swinging a sword around. One perceives the activity as spiritually significant, while the other perceives it as ... well... swinging a sword around for an hour. The difference, I think, lies in the perception by the first of the activity as having implications beyond the activity as such. The activity has transformative power, perhaps. Maybe the activity is perceived as an attempt to achieve mastery/perfection. Maybe the activity is perceived as an exercise in surmounting fatigue/boredom/pain.
There is often a quasi-Nietzschean aspect to assertions of spiritual significance for martial training -- "what does not destroy me makes me stronger." Perseverance in the face of adversity in one domain translates into an increased ability to persevere in the face of adversity in other domains.
The activity may also be perceived as being part of the construction of one's identity as a person. The "meaning of life" for a person is, in large part, a construction, based on the things and activities in which we invest ourselves (e.g., "I am an aikidoist" or "I am a Washington Senators fan" or "I listen to AC/DC"). The more an activity or thing is perceived as contributing to a person's identity as a person, the more that activity or thing is likely to be perceived or described as having spiritual significance.
Well, don't want to ramble on too long.