Why Trump's Wall is Not the Berlin Wall

I saw a legal blog comparing President Trump's wall across the Mexican border with the Berlin Wall. This was in the Volokh Conspiracy and framed as Trump's Plan to Force the Mexicans to Lock in Their Own People. So, why is this comparison wrong?

1) The Berlin Wall separated Germans from Germans. In the wake of World War II, Germany was divided into four sections managed by the US, England, France and the USSR. The first three were merged into a single government but the USSR-controlled region was kept separate. The Berlin Wall was built 12 years after the founding of East Germany. So it was separating Germans from Germans. Mexico and the US are separate countries with very different histories and cultures. Even the parts of the US that were previously owned by Mexico have been states for over a century.

2) The East Germans were prisoners in their own country. The Berlin Wall actually surrounded West Berlin which was deep in East Germany. Prior to the wall, Germans had escaped to the West by crossing into West Berlin then traveling to West Germany. The border between East and West Germanies was also closed. In contrast, Trump is insisting that Mexicans and South American refugees be prevented from crossing the US Mexican border but he in not insisting that Mexico's other borders or coast be closed.

3) It was death to cross the Berlin Wall. There were actually two walls with a killing ground in-between. Anyone seen in the killing ground was shot on sight. That's very different from the US/Mexican border.

This comes down to the question, does the US have the power to determine who enters it? If the answer is yes then the President is justified in taking steps to secure the border.

There is a substantial group who does not believe in borders and believes that anyone should be able to go anywhere with no attempt at assimilation. This will ultimately be self-destructive. Allowing multitudes of an-assimilated immigrants will eventually over-tax the country and destroy the institutions that made it desirable in the first place.