As Paul Johnson points out, if speakers don't state their
motivations we can't answer definitively for them. But it does appear to be
an attempt (not for the first time) for American "nativists"† to narrow the
definition of who is the "natural" ruling group for some part of the world.
Similar terms and ideas go back a long way, and were related to determining
just who was and was not properly "white." (The broadly accepted consensus
within "white" groups has changed over time and of course is still disputed
amongst various groups today.)
Robert P Baird, in "The invention of whiteness: the long history of a
dangerous idea" (The Guardian, 2021-04-20), writes:
Thanks to its role in facilitating slavery, whiteness in the US was often
defined in opposition to blackness, but between those two extremes was
room for tactical accommodations. In 1751, Benjamin Franklin could claim
that only the English and Saxons “make the principal Body of White People
on the Face of the Earth”, and nearly 80 years later, Ralph Waldo Emerson
would insist that the Irish, like the Chinese and the Native American,
were not caucasian. Over time, however, the definition of who counted as
culturally white expanded to include Catholics from southern Europe, the
Irish and even Jews, who for centuries had been seen as quintessential
outsiders.
This expansion of who is "white" was of course opposed by some, and
"Anglo-Saxon" became a way to distingish a more narrow definion of who
should belong to the ruling group from the expanding definition of "white."
Adam Serwer goes into detail about this, in "‘Anglo-Saxon’ Is What You Say
When ‘Whites Only’ Is Too Inclusive" (The Atlantic, 2021-04-21):
...it helps to understand that “Anglo-Saxon” is what you say when “whites
only” is simply too inclusive.
The Anglo-Saxonism to which I refer has little to do with the Germanic
peoples who settled in medieval England. Rather, it’s an archaic,
pseudoscientific intellectual trend that gained popularity during the
height of immigration from Eastern and Southern Europe to the United
States, at the turn of the 20th century. Nativists needed a way to
explain why these immigrants—Polish, Russian, Greek, Italian, and
Jewish—were distinct from earlier generations, and why their presence
posed a danger.
They settled on the idea that the original “native” American settlers
were descended from “the tribes that met under the oak-trees of old
Germany to make laws and choose chieftains,” as Francis Walker put it in
The Atlantic in 1893, and that the new immigrants lacked the
biological aptitude for democracy. Anglo-Saxon was a way to distinguish
genteel old-money types, such as nativist Republican Senator Henry Cabot
Lodge, from members of inferior races who had names such as, well,
McCarthy. The influential eugenicist Madison Grant insisted that the
Irish possessed an “unstable temperament” and a “lack of coordinating and
reasoning power.”
This eventually did lead to the "Aryan" distinction:
This belief that America’s “original” population was Anglo-Saxon, and
that the American way of life was threatened by the presence not just of
nonwhite people but of inferior, non-Anglo-Saxon (or “Nordic”) white
people, shaped the racist immigration-restriction laws of the early 20th
century. As historians have documented, it also influenced the ideology
of Nazi Germany. Translated into law, it produced such
horrifying artifacts as Virginia’s 1924 anti-miscegenation act, passed
with the aid of the eugenicist Anglo-Saxon Clubs. The law required all
babies to be classified as “white” or “colored” and made it a felony to
“misrepresent” your racial background. The Nazi jurists studying American
race laws in the 1930s thought such “one drop” rules were a bit
too strict.
However, the distinction as made in this case seems to be including groups
that would previously have been excluded from the "Anglo-Saxons" and were
certainly "non-Aryan" (by early 20th century definitions), such as those of
southern European descent. However, though the particular ethnic/regional
groups have changed (as they always do over time), the general exclusionary
intent and its justification has not:
...it’s clear that prominent Trumpist officials and intellectuals, some
of them descended from the very immigrant groups Anglo-Saxon was intended
to vilify, agree with some of the presumptions of Anglo-Saxonism. The
echo of the notion that, as Francis Walker wrote, non-Anglo-Saxons are
biologically incapable of “self-care and self-government” can be heard
regularly on outlets such as Fox News, where hosts like Tucker Carlson
argue that Democrats wish to “replace the current electorate,
the voters now casting ballots, with new people, more obedient voters
from the Third World.” This is biological determinism, but it’s also
simply false. The Republican Party is now led by the descendants of the
people Walker decried as incapable of self-government, people with
surnames like Giuliani and Pompeo, even as it launches these old
calumnies at a new generation of immigrants.
Both articles I've quoted here are well worth reading in full. If you're
short on time, Serwer's article in The Atlantic is the shorter
read and more directly relevant to this StackExchange question, but
Baird's article in The Guardian provides better detail and
historical perspective on the underlying motives and how this all came
about.
†"Nativist" here is what the descendants of certain North American
settlers called themselves; they are of course not the earlier natives of
North America whom they conquered and displaced.