how did hayko feel about his birthdays/life milestones/etc. that passed while he was with nick? was he able to enjoy them or did he see them as reminders of how long he'd been there?
Nick, despite Hayko's best efforts to celebrate quietly and alone/with Vlad, did manage to both find out his birthday (and some others like his law graduation date) and celebrate it with him. Or, firmly encourage him to celebrate together.
In that way, Hayko didn't just yearn for private celebrations of birthdays/milestones but started to... dread them a little. Nick wouldn't do anything terrible. But the cake, dinners, invasive gifts, irritatingly warm congratulations would have that tense undercurrent of performance and sour the mood for him (and sweeten it for Nick).
A domestic scene at Nick's place. Dessert and tea, made just the way he likes it. Expensive but out-of-place gifts. A designer tie. Sometimes, thoughtful but creepy gifts. A new pillow, because Nick "could tell that his has been bothering his neck". Shameless prying into his life, the date they're celebrating, its significance, information about his family... that Hayko would refuse to give.
It was a mockery. To Hayko.
To Nick? I think he just organically enjoyed it tbh, besides watching him squirm.