In short, the multiverse is a concept in theoretical science which posits that our known universe exists alongside a potentially infinite number of alternates. In the show, the quantum multiverse theory is adopted, in which a new universe is created when a diversion in events occurs, referred to as a nexus event.
As stated by miss minutes, the original multiverse was plagued by war across the realities in a way similiar to the avengers - pop in, steal an infinity stone, and be back to your own reality with none the wiser, but having completely changed the course of another reality.
Although vague, the presented information states that the sacred timeline was created, probably a single strand of the universe, to stop the multiverse war, most likely time travelers wreaking havoc or exploiting the past peoples with the advantage of knowledge and in the process creating new multiverses which is mentioned to be destablising (destablising as to what though is yet to be revealed).
The entire point of the TVA it seems is to prevent the creation of new multiverses by sticking to a single "sacred" timeline that is apparently for the greater good. If a diversion in events would occur a new parallel universe would be created. 283
This is reflected in the show as any time the TVA enter the timeline area they take great care to cover their tracks and return it to its original state so as to not trigger a multiverse splitting event.
As mentioned in one episode, destructive events take no bearing on the previous state of things and create something anew from its constituents - for example the destruction of a planet bears no memory on the state of its configuration (people, cities, plants) for its contents are essentially atoms. For example it would make no difference if you were at the bus stop or the library if the earth was destroyed - either way we would become atoms in the cosmos, and unless by some miracle of gravity and geometrics, creating of a new universe would be significanltly unlikely, since our atoms will only be absorbed by a neighbouring gravitional body such as the sun or jupiter in which case randomness or entropy would obscure all memory of the previous configuration.
Hence why the second variant is able to obscure herself in the midst of chaos as her presence does not really increase the chances of a multiverse by any significant amount.
So, to answer your question: How is it possible that if only a single universe is allowed there can be multiple variations of the same man?
So the TVA cannot stop the splitting of a universe into two but they can guide the events so that the result of a split recombines into the same outcome and resolves into a single universe once again, as long as the split was not large enough to be irreversible.
For example, if there was by some way of accessing true chance, you could effectively split the universe IF and only IF the outcomes of the dice throw were magnanimous in effect.
For example, thanos when he equips the five-stoned gauntlet, if he decided five fates for the nine realms - obliteration, relocation, preservation, and some others - each with a completely different outcome based on true chance that could not be recombined into the same result after taking effect - then you could effectively create five universes and five Thanos variants.
But the TVA being the TVA would not allow this to continue and would probably guide the entire galaxy in each universe which contained thanos and everyone else in the avengers show into a black hole in which case only the mass of galaxy - unchanged by the split of the universe into five - would be important and that all memory of the orientation of the galaxy would be lost forever, thereby forming the multiverses into a single universe again as the timelines again become identical, as the same mass had been added to the black hole in each case and the resulting effect was the same.
In a nutshell, yes there are multiverses created spontaneously to account for variants, however these multiverses are guided back to the original design or "scared timeline" for the universe by the TVA. As a result these multiverses recombine into the original scared timeline, which is easiest done at the time of the split, before the two parallels universes grow apart and become difficult to recombine to a shared and homogenous destiny (as the show demonstrates by not wasting any time after <PERSON> originally escapes).
At least that's my take. 283