What is the name of this kind of embroidery stitch:
Based on this encyclopedia of embroidery stitches, it looks like this is a "darning stitch", which is in the running stitch family.
The darning stitch is about making rows of straight running stitches near each other. The technique of darning is used to mend torn clothes, especially socks and looks like a woven patch.
Other sites corroborate this terminology.
The Wikipedia page for "running stitch" defines the darning stitch as:
Darning stitches are closely spaced parallel rows of running stitches used to fill or reinforce worn areas of a textile, or as decoration.