That's because there IS only one of each warframe. According to lore, each Tenno's power manifested differently. They channeled their unique strengths into the Warframe, their own ideas and talents molding the kind of Warframe that they would control.
At first, they turned to already-made Warframes that were simply 'close enough'; the Umbra warframes. They were prototypes; tests. Only a handful were ever used. They simply found Tenno who's particular 'void manifestations' were similar to the old, abandoned 'bio drones', and crammed them into them via transference. This is why Umbra warframes, the ones who have people inside them, are a rare and a lost secret; because according to lore found in The Sacrifice, they were from an older abandoned project and were ultimately just 'test subjects'.
However, once it became clear how useful the Warframes were, they began to craft Prime warframes as the pillars of strength. These did not have people inside them; they were made specifically as vessels for the Tenno, without a 'will' to animate them, so as better to fool the Tenno who slept in the second dream. As the war grew on, while the Tenno were winning, sometimes the Prime Warframes would take damage, and the beautiful Prime Warframes would have to be repaired...
...Thus, came the ordinary Warframes; mass-production models, cheaper to produce, and more replacable when damaged. Eventually, the Prime Warframes were kept only in Orokin Vaults, no longer used by the Tenno except for ceremonies and great need. That is why we all start out in Normal Warframes; because the Primes are lost in Orokin locations, where we left them long ago.
As for the Varients/Skins, its likely these models were designed as side-grades for the tenno to use; similar to the way that a military unit will change gear depending on his surroundings and objective, but might keep his skillset intact; Immortal skins might, lorewise, be for a mission that was a suicide run. Rhino's Palantine Armor? Likely a heavier-plated version for more intense firefights. Excalibur Prisma? A more energy-focused build for more Exalted goodness. Whether they were used or not (some lore points to some varients never being used) is simply because they were created for them, but the unit in question never was deployed on a mission that called for it.
The stats don't reflect this because DE doesn't want to have to balance that, and because it would bring up pay-to-win issues as far as stats go. In-game, its just skins. Lorewise, you look at me and tell me that Palatine isn't tankier than Rhino.
As for the cutscenes that feature warframes specifically, it is doubtful that they can be considered fully canon in their own right, but should be considered 'canon enough'. A warframe DID probably die in the Profit trailer; but they can't very well be Excalibur if your particular Tenno IS excalibur, right? But DE can't simply design a warframe and NOT give the players access to it. The community backlash would be too intense. Therefore, every warframe is available; but canonically, not every warframe is 'alive' today.
Keep in mind EVERY CHILD ON THE ZARIMAN 10-ZERO BECAME A TENNO. Every child. The Zariman was a COLONY SHIP on a mission. Colony ships easily hold thousands of people. THey had to have enough people to colonize an entirely new planet; and plenty of children to learn HOW to do it alongside the adults, so as to have a trained next-generation, in case colonization took longer than they thought.
There could easily be a THOUSAND Tenno lurking around the system, especially with the fact that there are multiple relays acting as meeting places. We are just one of the many.
Also worth noting, the players ability to switch warframes is NOT canon. That is gameplay, taking precedence over Lore. This is why the Operator was so distraught in The War Within, moaning piteously about how "I lost my Warframe!" It isn't just one in his collection; its his ONLY one. We, the player have variety, but the CHARACTER we play as does NOT.
Ordis might make a comment about Warframe specs, acquiring blueprints, and encourage us to build other Warframes, but if the Operator could so simply 'switch warframes' at will then the entirety of The War Within shouldn't have been a problem. In fact, every single Warframe 'fatality' that has taken place since the beginning of the war shouldn't have taken place if the Tenno can simply switch to a new one. The Tenno, now that they have 'awakened', MIGHT be able to do so; but its not normal, or usual. I personally believe that the Tenno retrieves the Warframe blueprints, builds them, and brings them to the Relays to give to a Tenno who has awoken but doesn't have a transference link.
Imagine the Tenno who went into the Long Sleep between the Old War and the Present Day, but who's Warframe was destroyed before they awoke? They would have stayed asleep until the Second Dream, when the tenno awoke properly.
When they woke up, they'd need a Warframe. That's why we're building them; we might borrow them, from time to time... but they're THEIRS.
It also seriously raises the questions of body dimorphism; how would you react when awakened, if you visited a relay, and see another tenno... 'wearing your old skin', so to speak? Imagine an Ember who visits a relay in tenno form, seeing the next Ember walking by in their warframe?