I would avoid trying to apply modern biology to a fantasy setting, as it make no kind of sense. Why should a fire giant an a frost giant not be able to interbreed? Physically, both are similar to humans only bigger (which is physically impossible in and of itself). And if they can freely interbreed, they are the same species, says science.
So, it makes more sense that it's the kind of magic they are connected to that makes them different, not biology.
Linnaean hierarchy is just putting an artificial classification system on what we find to give some order to it that makes some sense. I agree species is simply the 24 term, race is the 14 term and neither are meant to really be the exact match ups to scientific species usage in their D&D usage. The 14 race phrasing is more in the "race of man" sense than in the common White, Black, or Asian type of usage with everybody still the same species who can interbreed with viable offspring.
The 14 MM referenced the different races of the giants "At the highest level of the ordning, the races of the giants are also ranked according to status." while different types of elves for example are subraces. So giant types are distinguished in 5e between themselves at the level of elf versus dragonborn and not at the level of high elf versus drow.
Whether different types of giants can interbreed in 5e lore though is an unknown as far as I am aware with no even indirect references until we get to 24 goliaths being descended from giants (again I don't have the Bigby's expanded lore book and there could be other references), so up to that point it is unspecified like tieflings and gnomes. Whether you get reproductive half giants or sterile mule style hybrids or giants or nothing is not defined in the 5e core books that I see. It seems to be a DM choice situation.
In Volo's Guide to Monsters in the Giants expanded lore section it says: "In an age before human and elf, when all dragons were young, Annam the All-Father put the first giants upon the world. These giants were reflections of his divine offspring and also children of the world, birthed from the marrow of mountains, the hot blood of volcanoes, and the breath of hurricanes."
Can something birthed from the hot blood of volcanoes breed with something birthed from the breath of hurricanes? Maybe?
They each have a different racial patron deity even though many worship other gods. This can cut either way as you look at Moradin versus Correllon but also Correllon versus Lolth and Rillafane.
Giants are not just differentiated by magic and elemental abilities in 5e, their average size by type ranges significantly from 16' tall to 26' tall. Fire giants throughout D&D when specified (1e -3.5 explicitly, 5e gives no text description, relying solely on the art of them being broad and squat) are also morphologically similar to giant dwarves and not humans weighing significantly more than the taller frost giants.
5e MM giant size differences.
So for 5e a little less than 2/3rds ~ 65%? height increase to go from lower to higher average ends compared to the D&D humans just under 5' to well over 6' same species/race baseline a bit more than 20%? A human interbreeding with a giant (making them and goliaths and giants all the same scientific species?) to have goliath descendants would have a height differential of more than a 200% increase to get to the 16' tall hill giant. The 5e 24 PH goliath description just says descendants, it does not specify human interbreeding with giants though the 24 goliath PH description is similar to 24 tiefling with just saying tiefling and the 14 PH tielfing description specified human bloodline.
In the 5e Volo's Guide to Monsters it is interesting in the giants chapter there is no mention of goliaths and in the new races section that introduces Goliaths to 5e the one and a half pages of mostly lore on the goliaths never mentions giants at all and just talks about Goliaths as a big mountain habitat based race.
