If the Ivy League meant that those are the best, the most rigorous, and the most selective universities in America, then I would agree that Stanford should be a part of that group. However, while the Ivy League schools are known for "academic excellence and high selectivity in their admissions," they are not actually the best 8 universities in the country. According to US News, the top eight American universities are Princeton, MIT, Harvard, Stanford, Yale, University of Chicago, Johns Hopkins, and UPenn. Only 4 of these are Ivy League. According to Niche, the 8 most selective schools in the country are Harvard, Stanford, Princeton, CalTech, Yale, MIT, University of Chicago, and Columbia. Only 4 of these are Ivy League.
The Ivy League doesn't actually mean that these are the very top schools in the country (all though they are some of the best). So why are these schools grouped together? This article says that the Ivy League got its name because these schools had a tradition of growing ivy in the 1800s. Then the term "ivy" started being used for these schools in 1933 to refer to their football season. Then in 1935, "ivy" became synonymous with older east coast colleges. So it seems like Ivy League just means that these are old east coast colleges with a tradition of growing ivy, and they only happen to be among the top schools in the country.
I don't think Stanford would really fit in with this definition, though I agree with you that it is academically on par with the Ivy League, and it shouldn't be overlooked when ranking the best schools in the country.