I believe that, strictly speaking, Treasury tags are the longish ones of which one metallic end is inserted into the other hollow end to secure the documents, whereas India tags are the shorter ones that have metal parts like crossbars at each end of the ribbon. The former go through both punch-holes in the file-cover and sheets involved before being linked whilst each of the shorter ones go through only one of the holes
I cannot be certain, but my guess is that they took their names from the departments/buildings in which they were most commonly used. 'India' used as an adjective, as here, often referred to the East India Company, so perhaps tags of that particular type were commonly used in the company's HQ in London and/or in their offices on the sub-continent. By contrast, the Treasury may have tended to use the insertion-type for preference; perhaps their files were fatter!