(Originally written by altar)
Broadly speaking, metalcore and deathcore both come from punk rock. In its original meaning, "metalcore" was called "metallic hardcore" - punk bands who were taking influence from the crossover and mutual informing of punk and metal as extreme genres. Metallic hardcore added a caustic take on hardcore punk and "post-"hardcore (using the original definition of the term). The original metallic hardcore bands of the very early 90s include Integrity, Rorschach, Amenity, and Starkweather, with later acts like Cave In, Coalesce, and Converge adding other flavors of increased intensity and musical complexity (with the last of that list being a codifying band for "mathcore"). But the focus on breakdowns and the speed/aggression of hardcore punk remains (similar to fusion genres like grindcore). This isn't too surprising - crossover thrash in the mid-80s was basically an infusion of hardcore punk and thrash metal (e.g. Suicidal Tendencies), and "metallic hardcore" can kind of be seen as a similar offshoot or fusion.
Similarly, deathcore came from bands like Damonacy who were infusing the exploding and sonically intensifying death metal scene of the early 90s with the "metallic hardcore" approach (think Cryptopsy/Malignancy + punk, not Death's SBG + punk). "Deathcore" from the 90s might surprise listeners who are primarily familiar with the scene through modern bands. Deformity's Murder Within Sin is often considered one of the best examples of this kind of sound - others include Candiria's Surrealistic Madness and Day of Suffering's The Eternal Jihad.
Bands can be on the metal or punk side of the spectrum, and it's hard to pinpoint exactly when a metalcore/deathcore band becomes more one or the other. It gets nebulous, though generally speaking it's well-recognized that bands such as Cave In, Converge, The Dillinger Escape Plan, and Rorschach are much more on the "punk" side, whereas Deformity, Integrity, Ringworm, and Merauder are more on the "metal" side. The ability to tell the difference is up to your ears, and we use the defining line of "is it on Metal Archives?" just to provide a basic differentiation in discussions on the sub. It's not perfect as edge cases get lost, but it's good enough.
I'd argue that since the mid-2000s, both metalcore and deathcore have diverged so far from their origins that they're part of a scene distinct from both metal and punk. A lot of OG 90s metallic hardcore and deathcore bands are still very rooted in their parent genres. Bands like Integrity, early Cave In, Botch, 7 Angels 7 Plagues, early Converge, and Coalesce were capital-p Punk.... while Deformity, Mörser, and Day of Suffering came from a death metal background while incorporating those punk influences.
You start seeing a difference in the mid-2000s with a lot of new metalcore bands shifting into a new, melodic style that was strongly influenced by Gothenburg melodeath (e.g. At the Gates' Slaughter of the Soul) and developments in punk. Bands like Killswitch Engage, As I Lay Dying, Underoath, Protest the Hero, Parkway Drive, The Devil Wears Prada, and Vanna sound quite different from the 90s stuff, and this trend has only continued with time through bands like Code Orange. Deathcore underwent a similar change, especially with influences from djent and brutal death/slam metal, which is why lots of 2010s deathcore has a reputation for being extremely chug-heavy and focused on breakdowns even more than metallic hardcore.
It doesn't really matter if it's metal or not in the sense that something being "metal" isn't a mark of quality. If you like modern metalcore or deathcore, then great. There are plenty of excellent non-metal bands, and there are plenty of terrible metal bands. Lots of metalcore and deathcore are distinct from metal but certainly have a place among more "extreme" areas of music in general, which can cause listeners to become confused when they're used to think of every guitar-based form of heavy music as "metal".