I was curious about this as well. The paper you link is providing an extension of the binomial coefficients. Binomial[-1,-1] = 1 isn't the result of an "assumption" per se, but rather the choice of a particular approach to extending the coefficients (which are normally only defined for non-negative integers n, k). The paper does this by writing the coefficient formula in terms of gamma functions, and substituting a finite expression when the gamma function is infinite. Clearly, Knuth does not prefer this generalization.
The Knuth-coauthored text "Concrete Mathematics" takes a different approach, where instead of a substitution, the binomial formula takes its limiting value when the generalized gamma function form results in ∞/∞. So Knuth has
For any complex numbers z, w. It's not clear to me whether there is a further justification for this approach. I don't see a reference for this in the Concrete Mathematics text, but would be curious to learn more.
https://arxiv.org/pdf/1105.3689.pdf
cited by 32, proves Theorem 2.1 that gives Binomial[-1,-1] = Binomial[0,0] = 1. What exactly is the assumption on which they differ?
Whatever value you set this coeficient to, it's clear that not all of the familiar binomial properties
1. (n k) = (n-1 k-1) + (n-1 k)
2. (n 0) = 1
3. (n k) = (n n-k)
can remain true, as 1. gives 1 = (0 0) = (-1 -1) + (-1 0), and 2. makes this sum 0 + 1, while 3. would make it 1/2 + 1/2.