We have a tea set: a pot, a milk pitcher, a sugar container and another like-container with a lid. What is th?!

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It's for Honey, just use a spoon to pour it out.

It's NOT for jam or honey. Tea sets don't include containers for those two items.
My initial thought is that it is a second sugar bowl from a separate set if it is a British Tea service, Oriental tea sets usually do not have creamers.
If you purchased the service new, then there should have been a list of what each item was for with it.

It's not a slops bowl either- those are usually shallow unlidded bowls.
I have a fairly formal silver tea service, it's antique and still has all the accessories. This is what is in it.
We have a tea pot, coffee pot and a third pot that you put straight hot water in (it's for those who like weak tea).
There is a creamer, sugar bowl- ours is an unlidded one for sugar cubes and has tongs.
A slops bowl and strainer and a serving tray.

As a rule it is called a tea set if it includes cups and saucers, and if it does NOT include cups and saucers it is a tea service.

Probably its a container to hold the waste tea bags. This I found online for your reference

* teapot
* teacup and saucer
* sugar bowl
* milk pitcher also known as a creamer or jug

In addition, a formal tea service would include:

* coffee pot
* kettle with spirit light
* slop bowl or waste bowl
* tray

If it is an English tea set that pot would be for jam or preserves.