Illustrator

Enamel pin badges are usually between 15mm and 45mm with at least 0.2mm separating each colour. The backing card is usually around 50mm by 80mm. The metal part of the badge can be plated with various other metals to get the desired effect. 
There are two types of enamel pins, hard and soft. Soft enamel pins are cheaper to produce and are capable of having brighter colours but are less durable. You can still feel the raised details of the metal walls of pin. Hard enamel pins are more expensive and durable but cant have as vibrant colours. They have enough layers of paint so the surface is completely flat.
Colours can be picked using Pantone, a colour matching service used for product design and manufacturing.
​
Below are some badges I found on Instagram that I liked the look of, especially the ones with the backing cards included.

Animal Crossing already has a simple art style and lots of lovable characters that don't have a lot of colours in their designs, so there are a lot of things I could've chosen. The game also has a lot of dialogue, and many good quotes.