Butter does freeze solid if frozen long enough and as long as it does not have salt added.
The salt added to salted butter makes it harder for the butter to freeze solid.
The temperature at which butter will freeze is around 32 F or below.
It may take as long as 7 hours for the butter to freeze.
If the butter is salted butter it may not completely freeze solid but it will still freeze somewhat.
Butter does not have a single freezing point.
Instead butter actually has two sets of freezing processes.
The first is the water phase freezing.
If you tried to freeze salt water you'll find that water keeps "coming out of solution" and freezing until you hit a point called the "eutectic" point, where it then freezes solid.
And if you want to freeze about a pound of butter, it will take approximately 6-7 hours to freeze it completely.
Oils and fats have much slower freezing processes than water because of their molecular makeup.
Fat molecules, unlike water molecules, are not polar, so their attraction to each other is much weaker and causes them to move about (and crystallize) more slowly.