I mean, it’s not completely impossible since the doctor said his Omega traits are weak (and so are his pheromones). He probably didn’t go into heat very often, unlike other Omegas.
Just to share a little memory with you on a similar topic of forgetting “important” things…
I once forgot my middle name. Literally forgot it—as in I had absolutely no memory of what it was. I had to ask my parents what it was and their laughter still rings in my mind. ( ̄∇ ̄")
He definitely had informed the company. His colleagues know he has to see their ID to know who they are. But he doesn't told them again when they seem to forget or when new comer doesn't know because he thinks it’s not necessary for him to say it and he doesn't want to deal with people who seriously want him to remember them and people who want to test him whether he still can remember them if they dress or talk differently.
Spoiler! Spoiler!
He had been through that before so he doesn't want to deal with the same thing again.
About he doesn't forget that he is an omega but he was careless for once and he got pregnant.
He thought of taking birth control but he was a workaholic so work comes first and he forgot about it completely. Plus, he saw used condom, so he thought that should be ok.
I have same thing. I did told the boss but he said he will tell HR in case if something happens. He told me I don't have to tell others unless I really want to because most of time all have cards on their neck and I could distinct people by voices and lips. I do have trouble with new people but my boss knows so as my few coworker they just tells me their feature in advance if I tell them I couldn't difference his/her face.
I live in California so they have specific laws regarding disability they will not tell new coworkers and others unless I want to or they have complain about me. It depends on places back home everyone would laugh by 3rd week of my work since I would forget their faces even if I never tell them oh I can't remember faces or can't distinct faces. People notice the things we things we are hiding so well. My work speaks for itself as well. I would have harder time if I work in emergency sector as medical as it's detrimental to patients or liability of company but it's Computers zoom calls so they careless. I do love not knowing the faces tho because I don't have to deal with drama but our body is smart even with one senses doesn't work other part helps you navigate the world. Just yesterday I forgot the face of my new client. Since my body told me to smile at stranger because of they were too close. I smiled they complemented my other coworker they liked how she noticed me even she wasn't looking at me hehe. He will never know I smiled because my body just recognizes the presence of that body and I am good at voices too good. I can listen to pin drop in next room and their lips the way those move can tell who is who.

It's a genuine question for someone with the knowledge of this disability, because that's what it would be under: don't you have to inform your job about this when you are working with confidential information like Accounting? You can't recognize faces and you are in a management position and you need to know who is who because the information is important to clients and it can and will be a problem if you share information to the wrong person and legally that is a liability for the company. Again, genuine question who has more information about this because I would find that very concerning that no one at work knows.