We've had these things before - the pictures too - but I notice they weren't watermarked, so I've redone them just to make sure we're complying with Kew's requirements.
These measures seem to have a small following, and must admit I see them very rarely, but found these two this morning for £2. the pair, so couldn't leave them behind - I doubt that charity shops have any idea as to their purpose - probably lots of other folk too.
James Couper & Sons, Glasgow, produced and Registered three designs for Whisky measures - Rd. Nos. 110942, 110943 (both dated 20th August 1857), and 237141 (dated 14th December 1869) - as per the attached images from the Kew Archives.
The shorter of my two, from this morning, looks to be a good match for 110942 - apparently the capacity of Scottish Gills was smaller than their English counterpart - why I've no idea - and it's possible this measure is one Gill, but at the moment I don't have a cc measure to work out the capacity accurately, so not sure. This example of 110942, like some other designs of measures, has the indicator for the liquid level, in the form of notches cut around the outside of the rim - quite small oblique mitred cuts in pairs and as a single cut, both forms on the same rim - indicating that for this measure the whisky must be filled fully up to the rim, and unless you had some idea of what to look for, it's unlikely these marks would register as something of importance
For Couper's other Registration on the same date (110943), there would have been a similar form of mark or cut, on the body of the measure, to indicate the legal requirement of the minimum capacity.
However, the third of Couper's Registrations (237141 dated 14th December 1869) used a lead plug inserted into a hole drilled in the neck of the measure, to indicate the correct level of whisky required - and this lead plug was the subject of a patent taken out by Richardson of Stourbridge - although as mentioned the pieces were actually made by James Couper of Glasgow.
One of Richardsons sons went to work at James Couper & Sons, in Glasgow, where this later Registration was made, using the patented plug method, which was stamped by the Board of Trade inspector - with verification marks. I'm sure we've had an example of 237141 on the Board previously.
Turning to the other measure from this morning - no idea who made it or what capacity it holds - there's a large good quality circular polished depression underneath, with plenty of wear, but I can't seem to find a good match for the shape, and no idea as to whether Scottish or English.
I'd suggest second half C19, but not sure - these things apparently continued in use until c. 1930. If anyone can shed any light on the origins, or date, of this larger measure please do shout.
The capacity indicator on this larger one is in the form of a circular shallow mitre, cut around the upper part of the bulbous swelling of the neck - this cut can be seen in the photo.
The volume of this measure looks to be noticeably larger than the other one, and might even be one English Gill.
Feel free to comment or criticize etc.

Ref. 'Whisky Dispensers & Measures' - Brian Brooks - 2
There are two more pix which will be on the next post.