Depends how you view consciousness really. Whether you view it as a separate entity that can jump from body to body and yet- retain characteristics of itself. Or, whether it's simply an ability of the brain. Just like detecting sound is an ability of the ear.
So, after we die- of course there will be many billion more conscious beings born here. Before we were born, there were many million too. I guess it's down to belief on whether you believe those earlier incarnations or future ones were/ are 'you'.
I tend to think consciousness is simply a function of the brain. So, while I imagine there will be a huge amount of conscious beings born after I die, I don't personally believe that 'I' will have anything to do with them. They will be them- not me. Of course, it's nothing either of us can prove.
I agree that consciousness is certainly weird though. Horrific in a way. That we have the 'power' to become aware that we are in fact aware and trapped inside a meat prison. That feeling will also occur to countless others in the future.
The other thing though is- are newborn babies immediately aware of being conscious? What were your first existential thoughts? Do you have any clear memories even- the first 1 or 2 years of life? My guess is- no. I'd say- while babies are obviously alive and conscious of their own needs, they're not exactly self aware like older children and adults are. So- are we born fully conscious even- or, is it something that develops? Plus, if we've already lived so many millions of years- why are we so naive when we are born? We should all surely be genuises with all that accumulated knowledge and experience.
I guess it also depends on how you view self awareness/ consciousness. If you had been born as a crocodile say- do you think you'd have the same sense of self that you have as a human? If you were born as a jellyfish, would you even have many existential thoughts? It's not to claim animals are stupid but I'm not convinced they are as existentially aware as himans. If you were born as a different gender to different parents in a different country- how would that affect how you develop? Do you think you'd still have the same sense that you are you deep down?
That's an interesting idea though I think. Does your consciousness have a gender identity? How much of what we recognise as 'us' revolves around the physical body we inhabit?