What is the name of this prime number algorithm?
O'Neil [1] call this the "unfaithful sieve". It's much slower than the sieve of Eratosthenes.

For each prime $p$ you do work $\sim p/\log p$ and so the total number of divisions up to $x$ is roughly $x^2/(2\log^2 x)$ if you assume composites are free. (That's essentially true: they take at most $2\sqrt x/\log x$ divisions each for a total of at most $2x^{3/2}/\log x$ divisions.)

Divisions take longer than unit time, so the total bit complexity is about $O(x^2\log\log x/\log x)$.

[1] Melissa E. O’Neill, The Genuine Sieve of Eratosthenes