There is currently no way to do vertical stacking within the Kanban view like you can in the Gantt & Timeline views. To do so while keeping your data in Airtable you’d need to leverage external tools that provide the functionality you’re after. Considering Airtable allows this grouping in Gantt & Timeline views, it may just be a matter of time before they add it to the Kanban view.
That said, the Kanban view is IMHO one of the weakest of all the Airtable views, as the limited configuration options typically result in cards that are much too long & titles that are truncated to the point that it’s difficult to use. It hasn’t gotten any enhancements that I’m aware of since it was introduced, so you may be waiting awhile!