Even though the people of a species may be very visually similar in a lot of ways, there are going to be differences. You just need to decide what those are. It doesn't matter what species you look at in real life, be it birds, cats, fish, mice etc., no individual is perfectly identical to another. What the differences are can differ depending on which individual you compare to which. For example, any pet owner can easily tell apart their beloved pets, even if said pets are siblings of the same color and so on. I've had litters of kittens with, for example, multiple grey tabbies, but as they grow up they quickly become unique. Maybe one has slightly darker stripes, or one is a bit more filled out than another, or one has a slightly different shade of eyes or slightly shorter legs, or even just a different way of walking or a different sounding meow. In humans, take identical twins. It may be hard to distinguish between them when you first meet them, but usually anyone who knows them well can tell them apart with little trouble.
As the author, you decide what the traits of your bird people are. What traits of what types of bird they have, and what more humanoid traits they have, is all up to you. It's hard to give exact suggestions without knowing all this, but essentially: you get to pick what those small details are that set one person apart from another. Maybe their feathers come in different colors or patterns or length or shininess. Maybe some are smaller than others. Maybe they have varied eye colors. Maybe, as in real life situations where individuals are very visually similar, it's just a matter of how well you know them, and you can recognize the sound of their voice, the way they move, their style of dress (do they wear clothes? often graphic artists will use clothing and other context clues to help the reader distinguish between similar non-human characters), and other mannerisms/traits that make them unique to those who are familiar.
Edit: You comment that due to restrictions in anatomy for flight etc. there is less room for differences in body type, but that doesn't mean there aren't any at all. Look at real birds. Two pigeons hopping around in a park: pigeon number one is large and round (eaten too many bread crumbs!), whereas pigeon number two is smaller and very trim. There will always be some size differences (individuals who are slightly smaller or larger, often most obvious between the sexes) and weight differences (someone has been eating more or is more likely to gain weight than another) and build differences (even with the limitations for allowing flight etc., this would be the case. In real life this happens in animals as well as people, including birds. I have two male cats, both are well fed and muscular and even weigh about the same, but one is long and lean and the other looks more compact and buff). Age and sex can be a large factor in these differences, but they apply even among those with the same demographics.