Say you have a snake around your shoulders or holding a tiny animal and you wildshape, does the animal also merge into your form or is it dropped? Since it isn't technically equipment, but you can consider it worn.
According to the PHB yes.
PHB Pg. 67:
You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it.
Based on that if you own it and it is classified as equipment you can merge it. The sticky question is, does an animal classify as equipment.
PHB Pg. 141: under the Urchin background
Equipment: A small knife, a map of the city you grew up in, a pet mouse...
That indicates that a small animal can be classified as equipment, thus according to the PHB the mouse is "your equipment" and can be merged. I can see no reason why any other non-sentient animal you own wouldn't also fit this situation.
Or for that matter if you are evil and are down with slavery it could be argued that sentient creatures could be considered equipment. I wouldn't allow it but it is debatable.