That's because it's a different idea about reincarnation. The Hindus generally do believe in a soul or "atman" that reincarnates.

But the Buddhists teach about it in a different way. Buddha taught that we are confused about our own self nature. That we think there is something permanent that lasts from one moment to another - not as a theory but a kind of basic idea that underpins everything we do. He said that if you look clearly you'll see that there is nothing there of that sort.  That you never even realized quite how strongly you were clinging to this idea - and that if you could but see, you'd drop it instantly as if you found that all your life you've been holding a red hot coal in your hand.

Not that there is something there to be got rid of. To try to get rid of your sense of self is just going to make it stronger, reinforce the idea that there is some permanent self able to get rid of it :). You can never win with that approach :).

But he taught that at some point it will actually be something that you can see directly, dawn on you, that this self you thought was there all along - that actually - wasn't there, not as you thought it was.

Yet again is no value at all in affirming all this as  a creed, he taught also - he said "come and see" that you need to see it for yourself.

Most Buddhists haven't had any kind of understanding or realization of this. I know I haven't :). But to be a Buddnist - basically that's saying that you think there is some truth of that type to be seen. You maybe are doing an ordinary life just like anyone else, gardener, cook, programmer, musician or whatever.  It doesn't mean you drop everything to try to find this truth. But a kind of awareness of it as a direction and path, that's what makes you a Buddhist. A Buddhist is someone who is following the path of the Buddha, who taught that there is a truth of this sort to be discovered. With much else also part of the path such as opening out to compassion, and wisdom, and generosity and many other things. It's like there are other truths also that are connected together and the basic teachings on compassion, generosity etc and basic morality like not stealing, killing lying etc are the easiest to connect to for most people.

So - that's about this life. So - of course Buddha when he realized nirvana - he didn't disappear. He continued to teach for several decades. And talked about his past life, even previous past lives also, and taught people. Whatever he is talking about there, it is not anything to do with ceasing to exist as a person in the world in the normal sense.

So, in the same way, if you can continue from birth as a little baby all the way to an old person and death - why not also continue to another life? If you don't need an unchanging atman to be a person in this life, why would you need it to transition to another life?

So in that sense yes it is consistent with Buddhist teaching. But Buddha though he taught about his own previous lives, he said that trying to find out who you are in your own previous life, or next life, etc is not productive. He talked about his own previous lives, but warned about trying to figure out your own previous lives.

So questions like:

1. What am I?
2. How am I?
3. Am I?
4. Am I not?
5. Did I exist in the past?
6. Did I not exist in the past?
7. What was I in the past?
8. How was I in the past?
9. Having been what, did I become what in the past?
10. Shall I exist in future?
11. Shall I not exist in future?
12. What shall I be in future?
13. How shall I be in future?
14. Having been what, shall I become what in future?
15. Whence came this person?
16. Whither will he go?

That's the list of  "unwise reflections" in the Sabbasava-Sutta

So - the Tibetan system is kind of unusual. Most Buddhist traditions then you won't try to find your next or previous birth. And in Tibet also, ordinary 'Tibetans wouldn't know or expect to know who or what they were in a previous life.

It's not surprising you forget. Few people remember their early childhood and birth and almost no-one remembers being in their mother's womb. So why would you remember your previous life, separated not just by birth but also your death at the end of the previous life, and perhaps also the intermediate state which is described as being like the brightest lights imaginable and sounds louder than thunder.

So, it's in that context - that still some special teachers they think can be found again in their next life. But it's not like they are expected to remember their own previous lives like the Buddha. But as a child there is some memory and connection and when they die, if they have practiced well during their life, they can direct their next birth to the extent that they can leave hints and instructions to help people find them in their next rebirth.

So that's the basic idea and theory behind it, as best I understand it.

Other schools of Buddhism have much less emphasis on this. The Zen Buddhists especially hardly talk about rebirth at all. Yet they do have this basic idea of reincarnation, it is just that they are rather strongly following the recommendation of the Buddha to not follow it up and try to figure it out. So it's like a spectrum thing there.

And as there is no Buddhist creed there is no need at all to affirm a belief in past or previous lives. It is hard to practice as a Buddhist if you think this life is all that there is - because it is closing your mind off to possibilities as you can't prove that this life is all there is. And it's important practicing as a Buddhist to have an open mind because the path is to do with seeing the truth as it is, unfiltered by your preconceptions - that's where it is headed. So if you have a strong preconception that there are certain things you can never accept to be true, that closes your mind in a way that makes the Buddhist path rather harder to follow.

But if you keep an open mind about what happens when you die, that's good. That's the best actually, if you want to follow the path of the Buddha sincerely - better than having specific ideas about exactly what happens based on doctrine rather than your own direct understanding.

Though there is also  a place for faith in Buddhism where some people may come to thoroughly trust their teacher's understanding in a way that lets them quickly come to see certain things directly based on relying first on what their teacher says. That's a bit like the way that poetry or good music can sometimes lead you to connect to things in a way you hadn't realized you had the capability for before.

I am answering this, which I know is a tricky thing to answer in Buddhism because nobody else here seems to be presenting this. It may give you a rough first idea of how Tibetans think about it. If anyone else can give a better account, do say!