The simplest answer to your question is that he was largely admired for it, with this gushing Senate paean by Senator Smoot - best known to history as one of the authors of the infamous Smoot-Hawley tariff enacted a couple years afterwards - on the occasion of to one of its own being elevated to Vice President that expressed the view of the majority:

"Senator, now Vice President Curtis, began life amid primitive and humble surroundings. In his veins runs the blood of a Puritan Englishman, a French-Canadian, and an Indian maiden. That remarkable fusion brought forth the real American, whose career demonstrates the boundless possibilities in the land we love...We will feel more tolerant, more considerate, and more wise, as we look into his kindly eyes. We will profit by the example you have set for us."

There's actually a lot packed into that statement that needs to be deconstructed, but a much more simple reason for why Curtis faced essentially no discrimination is one he expressed himself:

"I'm one-eighth Kaw Indian and 100% Republican."

Both were indeed true; Curtis was not half Indian but 1/8th, and as I'll explain, he was also the embodiment of conservative Republicanism over the course of his 50 year career. Also, one editorial note: as I'm not versed in the debates over descriptive language here, I'm sticking with what my sources deliberately chose, which is Indian and mixed-blood, even though both are quite likely dated.

While he was indeed born in a log cabin, Curtis started off life with a bit more of a leg up than Smoot implies; Curtis' great-grandfather White Plume (who combined with his great-grandmother provided him his 1/8th Kansa/Kaw ancestry) had assisted Lewis and Clark on their expedition and when it came time to expropriate large parts of their land, the tribe seems to have fared somewhat better than others, although litigation over some of the land deals lasted from the 1950s all the way up through the 1980s. But while he was a self-made man whose first elected job was at 24 as his county's district attorney (where he shuttered Topeka's saloons as Kansas enacted Prohibition in the 1880s), given his family's land he was never in true poverty to the degree of many other Indians, and indeed was removed from the tribal roll around that time as he was mixed-blood and had not lived on tribal land for two years. He then was elected to the House, where he unsurprisingly ended up on the Committee for Indian Affairs and ultimately became its Chair, and later the Senate where in 1925 he became Majority Leader when Henry Cabot Lodge died.

This gets into an area where I've got little expertise - and where someone who does should feel free to weigh in - but a debate at the time was on the potential extinction of full blooded Indians and their land rights (along with defining who was a member of the tribe, tribal land, and government financial support) versus mixed-blooded Indians, where the latter could sell their land rights to whites under the proper circumstances.

This brought conflict between the two groups, and one explanation of Curtis' mixed legislative record reflects this: he fought for his own group, the mixed bloods who were willing to assimilate, but for others he had no hesitation promoting legislation and deals that were terrible - including vastly cheap mineral leases to Standard Oil for oil drilling on the lands of the Five Civilized Tribes. He did on occasion step up for some tribes (and was known as the "third Senator from Oklahoma" given how much he visited the reservations) when deals were grossly unfair, like the Kickapoos in Oklahoma and the Papgos in Arizona, and Curtis described the Five Tribes Indian Territory as “having about 400,000 people and something like 399,999 Democrats" so as in many things, partisan politics probably played a role in this.

All this made him popular among the Republican establishment, as Curtis reflected three things they respected: he pushed to lower the legal barriers on Indians selling land (allowing companies and land speculators to take advantage), he was living proof that Indians were entirely capable of becoming self-made men who could pull themselves up by their bootstraps (and thus importantly, government support could be reduced or eliminated), and while generally terrible with and uninformed about the details of public policy he was at the same time a master at 'whispering' in the House and later Senate - that is, being well liked among his peers on both sides of the aisle and getting deals done to produce legislation. One slightly absurd story about the first of these was while Chair of the Indian Affairs committee he had some of his own land sold out from under him to pay back taxes - and then leaned on the Commissioner of the Indian Agency to refuse to sign off on the certificate required to confirm that Curtis was considered competent to sell his mixed-blood land. This meant the tax sale was considered void, but made this member of the bar in good standing also the only Member of Congress to be declared incompetent to determine the proper use of his real property.

To the white establishment, he was a role model for what assimilated Indians could rise to; a 1925 pamphlet by the Indian Agency sent to all reservations - Curtis unsurprisingly encouraged it - was entitled "To the Indian Youth Everywhere" lauding Curtis' "hard work, persistence of honest purpose, careful attention to details..." and a few other bits of 1920s-era praise. To his own tribe, when he became Vice President there were celebrations ("eight Kaw full-bloods led a long procession to the tribe's dancing grove a mile and a half from Kaw City. After "five old chieftans" called a halt to the crooning and dancing, Curtis's sister apologized for her brother's absence and stated, "I want you to know that I—that we—are all very proud of Charles.")

However, for the Five Tribes and others, it was consistently a different story for decades:

"No man connected with Indian affairs is so heartily cursed in the Indian Territory as Curtis. . . . He is blamed for the appointment of a lot of carpetbaggers who are to blame for the present investigation and for the corruption that is found. ... To cite one case. A year ago Congressman Curtis held a council with the Creek Indians. He was there to further the interest of a select few friends regarding land and oil leases."

Last, his selection as Vice President is worth a few paragraphs.

Curtis had become a major player in the Republican party by 1920 and probably suggested Warren Harding during the backroom compromise at the convention, producing one of the truly terrible Presidents. However, he was not in the league of a William Borah or Henry Cabot Lodge. In 1924, Coolidge had solicited Borah (who he liked) to be VP in 1924 to solidify Western and liberal support - Robert La Folette had split with Republicans to run as a Progressive - but with Borah being somewhat non-committal, party elders tried to settle on someone else to run. Their inability to do so lead to one of the all time great convention lines ("I am going off to bed. The kind of man you are looking for as Vice President was crucified nineteen hundred years ago.") but they eventually settled on Curtis.

Except then, Andrew Mellon - probably the single most important party elder - suddenly reversed his objection to Borah. Coolidge promptly offered him the Vice Presidency, but Borah apparently took enough offense to Coolidge "meddling in the convention" ('cleaner' conventions were part of the Progressive legacy) to refuse the job - and with his good friend Borah refusing it on principle, Curtis did not dare accept either, resulting in Charles Dawes eventually receiving the nod.

In 1928, Curtis ran for President against Hoover - except that it was more a campaign for VP as it was widely acknowledged the convention was Hoover's to lose unless Coolidge either chose to run again or endorsed someone else. (Coolidge was lukewarm at best and somewhat cruel to Hoover during all this, and the two remained distant thereafter.) To shore up his flanks, Hoover sought out Borah's support - he'd have quite happily offered Borah the VP slot - and in exchange for several Progressive platform planks he got it. In turn, by then Borah had no interest whatsoever in leaving the Senate having become Chair of the Foreign Relations committee following Lodge's death in late 1924, wielding more isolationist influence over foreign policy for the next decade than any Secretary of State. Borah then recommended Curtis for the job over Channing Cox of Massachusetts (who he felt would drag down the ticket in the Midwest and West), and Curtis then was nominated and became Vice President.

In office, though, Curtis was largely inconsequential; he was included in cabinet meetings, but given his skillset was deal making rather than policy he contributed very little. (Possibly as a result, Curtis also left almost no primary source material.) The two minor incidents associated with his tenure were a protocol kerfuffle when his half-sister - Curtis was widowed - became his official hostess and was assigned far lower precedence than she should have been receiving, and a brief moment where he called in the Marines (by streetcar!) when the Bonus Army camped around the Capitol, only to be overruled by Hoover. Hoover thought about dumping him in 1932 given Curtis' strong support for Prohibition - which he felt would be the most important issue the Democrats could use against him in the campaign - but as Hoover discovered the hard way, the country cared far more about the depths of the Great Depression. Curtis toyed with running for the Senate again, but ended up spending the remaining 4 years of his life as a lobbyist in DC.

Sources: Borah (McKenna, 1961), Mixed-Bloods and Tribal Dissolution (Unrau, 1989), Herbert Hoover in the White House (Rappeleye, 2015), Hoover (Whyte, 2017)

Note: this does not answer your question, but this previous answer about Charles Curtis addresses his relationship with the Native American community and a few other aspects of his career.

• Why is Vice President Charles Curtis, a Native American, such an obscure figure in American History?
• Answered by /u/Vesploogie