Mazda MX5s manufactured between 1989 and 2005 have three different sizes of brakes:
The front calipers are different too. The Mk1 1.6 and 1.8 calipers look the same apart from the L and R indicating which side they are a different and they mount differently too. The Mk2.5 Sport or Big Brake caliper is completely different.
The Rear Calipers are the same for all cars apart from the Mk2.5 Big Brake.
All of these calipers and their respective brackets can be fitted to any MX5 or Roadster made between 1989 and 2005. So you could put Mk2.5 Big Brakes onto a Mk1 1.6 if you wanted to. You need to mount both the bracket and caliper together.
Here's a picture of all of the Caliper Brackets fitted to MX5s from 1989 to 2005. The fronts are on the right and the rears on the left. The Mk1 1.6 brackets are at the bottom, Mk1 1.8 and Mk2 in the centre and Mk2.5 Big Brake at the top.
Here are the front caliper brackets, they mount the calipers onto the hub. The largest bracket being for the Big Brake car. These allow the fitment of the larger discs.
The rear caliper brackets are all very similar. They are different sizes, as you can see and also have slightly different fitting bolts, clips and springs.
Here's a picture showing the key differences.
Mk 1 1.6 MX5s have the smallest brakes in the MX5 line-up. A common modification is to put calipers and brackets from a later 1.8 car.
You will need the caliper brackets, brake pad clips and springs, some caliper mounting bolts, the discs and appropriate pads. You can re-use your existing 1.6 calipers.