Which valve device is similar to a wye, but has an inlet and outlet of matching size combined with smaller outlets that take water from the main line?

Water Thief Appliance.

Explanation:

Water thief appliance is a device similar to a device called the wye. It is also called Three way valve. It is a small, gated manifold type appliance and it was developed in late 1930's.

It has an outlet and inlet of size that is matched and also has small outlet which takes water from the main line.

Hence we conclude that the answer is Water Thief Appliance.