Some organisms have a single hole that is used both for food intake and for waste excretion. Does the scientific community have a name for that hole? What is it called? Or is it simply called the mouth-anus?
This is sometimes referred to as the hypostome (most sources, though Wikipedia disagrees). It is usually muscular.
This is the muscular part of the tube leading to the gut which contracts in waves (the familiar peristalsis), pushing the meal into the gut. Sometimes undifferentiated from the mouth such as in the hydra where the mouth leads straight to the hollow interior.
Some sources have it that Planaria have just a pharynx, in others the structure the food enters before the pharynx is also referred to as the proboscis within a proboscis sheath, the "mouth" being referred to as the proboscis pore.
The intestine to cap it off.
This is referred to variously as the gut, the gastrovascular cavity, the digestive cavity and, of course the intestine.
If an organism has a single pore for both ingestion and excretion then it is known as incomplete digestive system. Primitive organisms like Platyhelminthes (flatworms), Ctenophora (comb jellies), and Cnidaria (coral, jelly fish, and sea anemones) has this type of digestive system. It contains only one opening, the "mouth" which also serves as "anus". Ingested material enters the mouth and passes through a hollow, tubular cavity called gastrovascular cavity. Cells within the cavity secrete digestive enzymes that break down the food. The food particles are engulfed by the cells lining the gastrovascular cavity.
There is no specific term for this orifice although there is this statement from the Wikipedia article of GI tract:
A through-gut (one with both mouth and anus) is thought to have evolved within the nephrozoan clade of Bilateria, after their ancestral ventral orifice (single, as in cnidarians and acoels; re-evolved in nephrozoans like flatworms) stretched antero-posteriorly, before the middle part of the stretch would get narrower and closed fully, leaving an anterior orifice (mouth) and a posterior orifice (anus plus genital opening) ...
Technically, digestive system of Sponges is also primitive, having pores all over the body called ostia where food is entered but excretion happens at a dorsal pore called osculum. (See image here)