The countries in the EU are not all economic equals. The differences between Eastern and Western Europe are quite stark, with economies sized from 7,000-2,700,000 euros. The GDP of Mexico is actually larger than Canada, and the GDP differences between Canada, Mexico, and the US are far less than many of the countries in the EU. With regards to per capita GDP, Mexico is 3 times less than the US. But many in the EU are the same when compared to the wealthiest member states. If you include Luxembourg, the differences are nearly double that of the US and Mexico.
It would also be incorrect to say that the cultures in the EU are all similar. There are many different cultures, as different as the US and Mexico. Additionally, there are many different cultures across the United States already.
Our welfare system does not grant all immigrants full benefits either. Only citizens get most welfare benefits, not even legal permanent residents.
A system where immigrants from Mexico can live and work legally in the United States (but not become instant citizens) similar to the process in the EU would be a major improvement in our current immigration system, in my opinion.