A migrant is simply someone who moves from one place to another, in this context from one country to another. The legality of said move is irrelevant, and they are additional people that need housing and services. The number of people who have moved to Canada, through an unending series of special programs and exemptions and visa waivers and programs is colossal.
But if we must talk about illegals, yes Canada has an enormous number of people who technically aren't allowed to be in country. They came here legally -- given that this government is so set on not being racist that they removed virtually all checks and restrictions -- and just...didn't leave, for instance it's incredibly common for legal temporary residents to bring their parents here to "visit", where visit means live here. And anyone thinking the five million with expiring visas are going to leave is deluded.
As to housing, it basically hit peak insanity in 2016 on its natural curve and stagnated through to 2020 (literally - 2020 was slightly below 2016, inflation adjusted). So this government juiced every immigration channel to the max, leading to a massive inflation in housing. Yes, it was immigration that caused that, as something where demand closely matches supply is going to be massively out of balance when suddenly demand vastly exceeds supply over a short period of time.
One of the parent posts lamented people "blaming immigrants". I don't blame immigrants. I do blame immigration, however, and a government that turned a blind eye to the catastrophic effects, buying as much time as they can by just telling everyone that it's racist to dare question mass migration or its effects. All of Africa, South Asia and the Middle East can't just move to Western countries. That isn't rational or sustainable.
"targets almost exclusively wealthier / skilled people"
Have you not updated your priors since 2003? Wealthier/skilled people make up a very small minority of Canada's new residents. An absolutely tiny proportion. Which is a big reason Canada has had multiple years of GDP/capita contraction. We have millions of doordash drivers and Tim Hortons workers imported from abroad.
https://dennisforbes.ca/articles/10000_brainiacs.html