First off, the "dreamers" are unlawfully present. They are fully aware that they are not legal immigrants.
Secondly, DACA stands for Deferred Action for Childhood Arrivals (and DAPA is Deferred Action for Parents of Americans and Lawful Permanent Residents) aren't really amnesties. Neither do they change the status from being unlawfully present to lawfully present. They merely "defer" any deportation.
DACA and DAPA recipients are still unlawfully present and would need to leave the country for the required period of deportation to even consider getting legal residency under current law.
On the other hand, Windrush refers to people who were citizens of colonies or commonwealth countries. The picture of the British Empire Passports gives you a clue to the problem. These people were under the impression that they were British Citizens, which they were up until 1971.
Wouldn't you think you were British if your passport said "British Passport"? Or said you were a "British Subject"?
First part of the changes were some of the places Windrush immigrants came from gained independence. The Second part was The The Immigration Act 1971, which ironically was also partly passed to legally clarify the rights of Commonwealth citizens within the United Kingdom in preparation for future membership of the European Communities which the UK became a member state on 1 January 1973 which gave new automatic rights to EC member state citizens.
So, it's amusing this is coming in tandem with Brexit which is now removing the right of residency for EU member state citizens. Anyway...
A closer analogy for people in the US would be an overzealous ICE (Immigration and Customs Enforcement) going after people who were from Puerto Rico, Guam, Northern Mariana Islands, and US Virgin Islands since those are territories, not states. The Philippines, Micronesia, the Marshall Islands and Palau are former territories which gained independence from the US. People from American Samoa can visit the US without a visa, but they need to go through the naturalisation process to become citizens.