I don't think this is universally true; if you look at the portraits by Hans Holbein the Younger for example it starts to become very obvious that these people all look different. Some people have rather large noses, some people have rather wide or round or skinny cheekbones, some people have the most absurd tiny eyes, and so on and so forth. This is most obvious when you look at his male portraits; it's a little less visible in his female portraits but it's present. Bear in mind also that all these people are wearing very similar clothing and formal robes; because we are used to using very visible differences in clothing habits, hairstyles and dress to tell people apart(think about how recognizable some of your friends are simply by their haircut or the way they dress) and strongly expect old portraits to be idealized(not that they weren't; but a greater or lesser degree of idealization is always a intentional choice on the part of the artist) we might also be less sensitive to subtle distinctions in their appearance. Indeed, once we look more closely at the portraits of his wives in sequence, salient differences emerge in the length of their faces, their noses, how deep-set their eyes are, the prominence of their chin and so on and so forth. A useful point of comparison is the rather nasty racist joke that 'all Asians look alike"; in the same way we tend to mentally categorize all people in 16th century paintings as looking alike. This extends to the clothing; the fact is that a typical modern viewer is going to be less sensitive to the vagaries of 16th century fashion and less sensitive(in a world where virtually everyone wears cheap and affordable comfortable cotton or synthetic fabrics on a regular basis) to the different textures and qualities of linen, silk, taffeta, velvet, or brocade is probably going to not recognize subtle differences in dress that would be glaringly visible to a 16th century viewer-look at for example how the portrait of Jane Seymour you probably saw shows a much wider collar than the portrait of Anne of Cleves, or how much more loosely and widely cut and fabric-intensive the clothes of Anne of Cleves look.

You have to remember that portraiture is a very formal event. It takes hours for an artist to paint a portrait and they were quite expensive. People would dress up in their finest clothing to pose for the portrait and put on their most serious expressions (smiling was not considered professional). For this reason, when looking at old portraits though our modern cultural eyes, we see them as serious and almost Orwellian, though this may not be the case.

So I'm not an art historian — I should say that up front. But you're not the only one to notice that this is the case. The artist David Hockney has wondered this himself, as both a practitioner and a viewer of art. Hockney has noted that there is an apparently huge jump in portrait quality — that for hundreds of years it is as you described, not very distinct, not very detailed, not very photo-realistic. And then out of nowhere comes a school of very very realistic painting (e.g. Jan van Eyck, Caravaggio, Velázquez) that looks totally different — damned near photographic.

Hockney, with the assistance of a physicist named Charles Falco, has put forward a theory that this change came about through the use of optical technologies, including lenses and mirrors, that allowed these masters to aid their painting by projecting the image of their subject onto the canvas. In other words, they "traced" some of it. He has found a lot of interesting evidence in favor of it, including distortions in paintings that are exactly the kinds of distortions you'd get if you were using lenses or mirrors, and by looking at the original sketches he says you can see evidence that they were tracing as opposed to free-handing (the strokes have a very different character to them). He also, as an artist himself, took pains to try and replicate these technologies. He would be the first to emphasize that this is not "easy" work — but it can help you anchor a much more realistic portrait than without it.

I found Hockney's argument somewhat compelling from a history of technology standpoint — the idea that they wouldn't have used these kinds of tools seems idealistic to me, and the fact that they would keep these sorts of things secret also is in line with other early modern optical practices (e.g. Galileo).

The book on this is David Hockney's Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters. It's a beautiful, fascinating, unusual book. It's a coffee table book size but it's making a real scholarly argument — it just needs the visual space to do it.

(There's a movie that just came out, Tim's Vermeer, which is along very similar lines. I haven't seen it yet and can't speak to it.)

My understanding is that most art historians are not interested in these arguments, but how much of that is because there is genuine dispute about the Hockney thesis, or because it disrupts their standard modes of analysis a bit too much, I don't know. I found it pretty interesting, though, and I suspect that in a few more "generations" of art historians there will be more acknowledgment about these kinds of possibilities. I think Hockney feels that the present discipline of art history is too invested in the idea of these guys being "pure" painters and is just unwilling to even talk about this sort of thing. Again, I'm an historian of science and technology, not art, so I can't speak to the latter with any authority. But I do find the argument interesting.