This is an interesting question, but even before I get to some data...
Are they apples to apples? How are the Canadian and American immigration landscapes similar and different?
Right from the start I would say the answer is no, it's definitely not an apples to apples comparison simply based on geography. Given the economic situations for a number of Latin American countries below the United States, there is certainly more immigration (both legal and illegal) to the United States than there is to Canada.
Taking a look at the Migration Policy Institute's map (data from 2019), you can see that all told there's a pretty big discrepancy between the total number of global migrants in each country...
- Canada: 8.05 million
- United States: 50.63 million
And then taking immigrants from Mexico for example...
- Migrants in Canada from Mexico: 87 thousand
- Migrants in United States from Mexico: 10.85 million
Here are some sources and interactive visualizations that speak pretty directly to sheer volume of immigrants and emigrants in different countries across the world (including the US and Canada)...
This Pew Research article also has some interesting data on citizens' perspectives on immigration across a number of countries (including Canada and the United States).
I'm more concerned with information about illegal immigration, but general information greatly appreciated as well.
Just by its very nature, it's harder to find concrete data on illegal immigration. However I did find at least one source for each country that tries to estimate it...
- Canada: While there are no accurate figures representing the number or composition of undocumented migrant population in Canada, estimates range between 20,000 and 500,000 persons. (CIMM)
- United States: ~11.5 million in February 2022 (CIS, see "Figure 1. DHS and CIS Estimates...")
More broadly, the United States' state department claims...
The United States hosts more immigrants than any other country, with more than one million people arriving every year as permanent legal residents, asylum-seekers and refugees, and in other immigration categories. In addition, hundreds of thousands of foreign-born individuals become naturalized U.S. citizens every year. The total across these categories, coupled with the United States’ openness to people of all races, ethnicities, and religions, makes the United States the world’s most generous country for immigration.
Assuming this claim and the other data points shared above are at least mostly accurate, I think they give a pretty clear picture of the differences in the landscape. Without taking sides I feel it's fair to say that given the significant difference in volume and more specifically illegal immigration, it certainly makes sense that the topic is more contentious in the United States. That being said, I think the two governments certainly do work together on immigration to some extent and both work with Latin American governments on immigration policy.
As noted by the commenters, this could really be a whole thesis but I hope I've provided a good foundation and some starting points. Here are a few more sources and articles I found interesting (the first three are focused on comparing policy)...