When you pick up a cow using the halter, it replaces the halter with a different halter containing the cow rather than storing the cow in the existing halter.

This means that when you pick up a cow, the halter is treated as a new item and moves from your hotbar to the first available inventory slot.

If your inventory is full, the halter drops on the ground.

Edit: This only happens when picking up a cow as a mob, rather than from a cow stall.