I'm sorry but I think you know this isn't about your insecurity, not even slightly. You're involved with a man who keeps you a a secret. He won't even tell his parents, never mind his friends or his kids. You're not meeting them as Partner. He even says on social media he's single.
He's 40 years old. Not a teenager. He's been separated for years.
His parents are spending his birthday with him as they think he will be alone.
The truth is he's not committed to a relationship with you. His actions prove it. I guess he's getting laid and that's all what he wants out of it, but doesn't want you involved in his life in any way shape nor formal his partner..
The longer you allow yourself to be treated like this by him, the worse you're going to feel about yourself.
Tell him that when he's ready to have a relationship to call you. But you. Can't continue with a man who denies your existence and continues to deny it after a year together.
Then walk away and if he comes back and decides to be open, all good. If not, then you have your answer, he never wanted a committed relationship with you. Tryto accept that's about him not you.
Out of curiousity. How often do you see him exactly? His behaviour is totally abnormal, so I also Suspect there's more going on in his life than he's telling you.