4e ensured that every setting had a context for Dragonborn and Tiefling. And players complained that the settings were being retconned. 5e has a light touch that relies on the DM to make sense of these species. And players complained that there is not enough context. I assume these are different players complaining in different play style camps.
With regard to setting, I prefer the light touch with suggestive description for inspiration but that allows the DM to make the setting ones own.
The Players Handbook has four species beyond old school. Awsimar, Tiefling, Dragonborn, and Goliath. The setting assumptions for each differ, albeit all descriptions are versatile. It is the job of the DM to decide how to integrate them into an ongoing setting that had not yet encountered them.
Awsimar and Tiefling are opposite each other, Celestial and Fiend, but both are Astrals with moreorless the same story. They are born sporadically into any Humanoid species, into any family in any culture. The Astrals lack their own cultures, besides the ones they happen to grow up into. These are Astral alignment magic becoming Material souls of flesh and blood. Notably, they are only born to Humanoids, thus the alignment magic seems to respond to Humanoid alignment behavior. As DM, I have the birth of each one be ominous, a sign indicating that the overall weight of alignment actions of a community has recently shifted toward Good or Evil respectively.
Dragons reproduce Dragonborn magically. These are Dragons imbued with a Humanoid soul. I am unsure how to characterize Dragons. At the moment I understand them as personifications of the Elemental Chaos and the roiling potential elemental energies there. Is some sense they are their breath weapon taking on a form of life. Dragons are an ancient, even primordial, creature in most D&D settings. In any case, Dragonborn are Dragons with a Humanoid soul.
What makes Humanoids unique is ones soul exists simultaneously in every level of existence, especially Material, Ethereal, and Astral. For example, the difference between a Fey Eladrin and a Humanoid eladrin is, the Fey lacks an Astral soul. The Fey soul is a kind of Positive Material soul with an Ethereal soul influence. Being Humanoid has pragmatic characteristics, including a Material body, capacity of speech, and the ability to learn and form cultures. But there is also a mysterious mystical microcosmic aspect of a Humanoid soul that fascinates other creature types.
Dragons create Dragonborn because of the multiversal interconnectivity of a Humanoid soul. Dragons helped form the Material Plane, and want to be part of how it develops. Dragonborn individuals can appear anywhere Dragons are. But there is also a sense of native Dragonborn cultures that include diverse species of Dragons as ancestors. So even in a single household parents can reproduce children each exhibiting the atavistic traits of a different species of Dragon. Such pan-dragon cultures would be noticeable. If characters were unaware of the Dragonborn, it is because their cultures are somewhere else on the planet. In Oerth, Dragon Island is a homeland. In Toril, Tymander is a homeland.
Goliaths are in a similar situation as Dragonborn but with ancient cultures that survive in environments that are too harsh for Human communities. They live alongside Giants. Goliaths are often reputed to be "half giants", but actually are Giants who gained Humanoid souls during unclear circumstances.
