from what i gather, this has more to do with due process than anything. and while I see where the court ruled aliens have the rights guaranteed in the constitution, this isn't what they are referring to. I'm no constitutional scholar but reading what you posted, it's clear the link is specific to due process and a right to a fair trial.
for one, they cannot purchase or lawfully possess firearms, vote, and are restricted to countless other things
edit for context:
In 1903, the Court in the Japanese Immigrant Case reviewed the legality of deporting an alien who had lawfully entered the United States, clarifying that an alien who has entered the country, and has become subject in all respects to its jurisdiction, and a part of its population could not be deported without an opportunity to be heard upon the questions involving his right to be and remain in the United States.1 In the decades that followed, the Supreme Court maintained the notion that once an alien lawfully enters and resides in this country he becomes invested with the rights guaranteed by the Constitution to all people within our borders.2
Eventually, the Supreme Court extended these constitutional protections to all aliens within the United States, including those who entered unlawfully, declaring that aliens who have once passed through our gates, even illegally, may be expelled only after proceedings conforming to traditional standards of fairness encompassed in due process of law.3 The Court reasoned that aliens physically present in the United States, regardless of their legal status, are recognized as persons guaranteed due process of law by the Fifth and Fourteenth Amendments.
)
