It doesn't sound like especially smart behaviour, because  it is very common. Many birds are fascinated by their own reflections and will fight them bitterly, sometimes till the shiny surface is blood-spattered. (If that happens to a shiny surface of yours, the best way to stop it is to smear it with a bit of soap or similarly harmless material, to reduce the reflectivity.)

Your little bird seems to be a less physically pugnacious species or gender, vainly trying to elicit appropriate socialising behaviour from its own reflection.

I think that territorial instinct is most important here. The reflection is just like 'him' and is a threat to mating since the size and coloration are the same.

This is a test of self awareness which some species (dolphins, elephants, some apes and human females) pass and some don't (most other creatures and human males) However some Parrots have passed it. It tests whether the creature thinks the reflection is of another individual, like your bird, or whether it recognises the reflection as being itself. Therefore the answer is No.

Obviously, as any Budgie owner will tell you, birds do react (amorously or aggressively) to their own reflections, but as the bird's eyes are on the sides of it's head, the straight down look would tend to show it the edges of the mirror, not it's own reflection

Perhaps you lot can see something I can't in the video which suggests recognition, and it's very likely that your cars are newer or cleaner than mine, but if I saw that behaviour on my wing mirror, I would just assume the bird was snacking on the spiders which usually live in the mirror casing and appear round the edges of the mirror.

Of course, if an ornithologist amongst you assures me that it's a bird which only eats seeds, and never insects (or spiders), then I will reconsider.

One of my parrots had a sexual relationship with his mirror. Actualy, with every mirror in the house or any other reflecting surface he happened to run into. He fed it, talked to it, did all those courtship things... The other one had a more sedate relationship with mirrors, he liked my left forefinger nail better, he was the picky one... :)