And it would be the height of irony to call buddhism a vedic religion, as it along with other sramana movements (such as jainism or ajivika) of the time precisely rejected the vedas.
It's a lot more complicated than that...
Vedas don't have any authority in buddhism, just like in other shramana religions, and seem their claim to traditional authority is directly criticized in actual buddhist scripture - literally that this tradition is blind men leading the blind (itself possibly an Upanishadic reference). No doubt there's a decent amount of complication in the matter of various lines of theological influences going between the religions as one would expect, but how is that any justification for calling buddhism a vedic religion? Hindu authors have commonly identified precisely Buddhism, Jainism, Ajivika and the Charvakas, in other words the shramana traditions as being nastika on exactly the basis of their rejection of vedic authority. Now if you'd call it Dharmic religions, sure that works - though is a bit of a strange euphemism in case of Japan where clearly one can only be talking about Buddhism, so why define it in any plurality like "dharmic religions"?
http://www.accesstoinsight.org/tipitaka/mn/mn.095x.than.html
" Suppose there were a row of blind men, each holding on to the one in front of him: the first one doesn't see, the middle one doesn't see, the last one doesn't see. In the same way, the statement of the brahmans turns out to be a row of blind men, as it were: the first one doesn't see, the middle one doesn't see, the last one doesn't see. So what do you think, Bharadvaja: this being the case, doesn't the conviction of the brahmans turn out to be groundless?"