How to get Board detail (name or id) for board column in case of shared area path between 2 boards ?

bizdevaccts we have to do it in multiple steps.

For example, work item here exist in two boards ("ProjectA Team" and in "MultiTeamDemo") , in same column Active. (it can be in different columns as well similar to yours)
https://dev.azure.com/chamindac/MultiTeamDemo/_apis/wit/workItems/2115

Now when list borads for team "MultiTeamDemo" with https://dev.azure.com/chamindac/MultiTeamDemo/_apis/work/boards
I can get the board ids. Then I get the board id of my stories board for team "MultiTeamDemo"

Then getting that board information with https://dev.azure.com/chamindac/MultiTeamDemo/_apis/work/boards/656164f9-01b2-46d0-bf13-f15e36334f13  (or you can use url for stories board returned by https://dev.azure.com/chamindac/MultiTeamDemo/_apis/work/boards) gives the refernece name WEF_FF38659E9F984B12B955413C347F6B1D_Kanban.Column

same way for  "ProjectA Team" team using https://dev.azure.com/chamindac/MultiTeamDemo/ProjectA%20Team/_apis/work/boards can obtain the boad ids

the board has the reference name  WEF_133F1CE23A904B95BFBD37CD9968BAB0_Kanban.Column in fieds

https://dev.azure.com/chamindac/MultiTeamDemo/ProjectA%20Team/_apis/work/boards/c609a8d9-9d84-4ec3-acd3-7c881988e8ab (or you can use url for stories board returned by https://dev.azure.com/chamindac/MultiTeamDemo/ProjectA%20Team/_apis/work/boards)

So it is not straight forward. What you have to do is

1. List all boards for your teams in the ADO project (including the boads of the team project (root team)), and filter for stories boards and obtain ids of those boards.
2. Then query each board with id and obtain the fields.columnField.referenceName, so you can have list (table) of board IDs and its column reference.
   
   This will give you a map between column reference name and the board id.