OK.... So what you seem to have there is some past owners DIY replacement for an Indicator Switch....
There are aspects of this installation which are confusing me, which makes me wonder if it is a Saloon to Convertible conversion.
Possibly even converted from LHD to RHD...
Image mm4. The DB10 Relay is marked as 8/61, so August 1961 production from Lucas. I therefore imaging the car is later in1961.
However....
Image mm1. This show the DB10 on the LH/Passenger inner wing, but it should be on the Drivers side...????
Perhaps it used to be a LHD????
Look under the front floor carpets, is the brake Master Cylinder under the LH or RH front carpet??
The Reinforcing Plates have holes in for the old Lucas Trafficator Switch, used up until the end of SII.
This plate does not have the hole when fitted to post 1956, S3 and S5 cars, because they had the Steering Column Stalk
As seen in this image of a 1960 car found online:

- 1960 layout.JPG (51.35 KiB) Viewed 6762 times
I assume your car hasn't got an Indicator Stalk.
Please tell us what you know about the car, including is it a split screen or not, what is the prefix on the Vehicle ID Plate near the Voltage Regulator.
We don't need the actual numbers at the end of it. It'll be something like MA*/*/*...
A photo of the engine bay like this, would be helpful.
The Red Ring is where the DB10 should be, along with the actual Flasher Unit. The DB10 is not the Flasher Unit, it only controls what lamps are flashing.

- DB10 Engine Bay.JPG (100.26 KiB) Viewed 6762 times
This car appears to have some very unique aspects, far greater than a mere unidentified switch you initially asked about.
Here's a clear version of what the connections are on the DB10.
This illustrates the Brake Pedal pressed, with Right Indicator Flashing.
if the Indicators are off, the un-powered Relay routes the Brake Light supply on 5 through to 3 & 7.

- DB10 Colour v3A.jpg (672.36 KiB) Viewed 6749 times