Connecting a Makita jigsaw (4351) to dry vacuum cleaner (Dyson)
The main problem with using something like a Dyson or any other "floor vac" is that their filters (and warranty) are not made for sucking up a stream of sawdust or other "shop dust".  I imagine you can, and it should work fine, but you could clog up a comparatively expensive filter in a short time.

A "shop vac" has large filters made for exactly this type of dust and the filters are pretty cheap and easy to replace.  Some shop vacs will also accept bags for certain very fine dust that would quickly clog a pleated filter.

Your other issue is that attachments for power tools are going to be made for the standard sizes of shop vac attachments.  The only thing the Dyson has to connect to is its own included tools so it might not fit anything else without a clunky adapter.