Is plain flour and all purpose flour the same thing?

Yes, to an extent. "All purpose flour" is a term used in America. While "plain flower" is a term pertaining to the Europe region. In cooking, they are the same thing. However, if you are looking to order some via mail "All purpose flour" is sifted, while "plain flour" is not. So, you might be confronted with an issue in that specific circumstance(e.g. the flour would become condensed.)

In the U.S, yes. Plain flour = All Purpose Flour.

You can see the back of the package for protein content.

Here is the approximate protein content of all the common types of flour:

Bread Flour: 14 - 16%
All-Purpose (AP) Flour: 10 - 12%
Pastry Flour: 9%
Cake Flour: 7-8%

I copied from this site.
What's the Difference? Cake Flour, Pastry Flour, All-Purpose Flour, and Bread Flour | The Kitchn

In Asia, usually used cake flour as plaing flour , which contains under 8.5% of protein.