What's the name of the approximation $(1+x)^n \approx 1 + xn$?

I would just call it the first order truncation of the Binomial series. If you want more terms of the series, then it's given by $$(1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \mathcal{O}(x^4)$$ for the full series, you can visit the link I provided.

You may also be interested in Bernoulli's inequality

I would say it comes from the Bernoulli inequality. You can read it up on Wiki