Sometimes people ask him if he is enlightened and he says he isn't.  You don't have to be enlightened to be a recognized rebirth in Tibetan Buddhism. It's like an inspiration carried from one life to the other. 

After all we are all rebirths, according to Buddhist teaching. There are many kinds of "tulku" in Tibetan teachings and the simplest is just someone who was a human in their previous life.

Though Buddha warned against trying to find your previous and next rebirths as unwise reflections on the Buddhist path. So most ordinary Tibetans would not know or want to know their past lives.

When Tibetans say he is the embodiment of Avelokiteshvara - that often gets misunderstood as if they are saying he is a Buddha. Yes in a sense, but in the same way that everyone who shows a moment of compassion is an embodiment of Avelokiteshvara. It's like Buddhahood shines through all of us. It's within that context that they say that.

And then - in the Tibetan tradition there's also the idea of a guru and pupil, which you don't have in quite the same way in all the Buddhist traditions.

There the idea is that you can relate to anyone actually as the Buddha. It's much more to do with your connection to them than intrinsic to them. The first person the historical Buddha met after he became enlightened was Upaka, who impressed by his serene appearance asked who was his teacher and what he had realized. When he answered that he had no teacher and that he had reached Nirvana, Upaka said "it may be so, friend" and went another way.

So you can't say that Upaka really met the Buddha in the sense of an enlightened person at that point.

Well it can work the other way too, you can relate to someone as a Buddha that to everyone else seems an ordinary person. Even animals, birds, inanimate things, can carry the inspiration of awakening, opennness, They can connect you to the truth, ground you.

It has to come from outside of your ordinary self in some way. Because otherwise it is just "same old". Usually that's inspiration of another person, but it can also be something inanimate. So that's the idea behind the guru  / pupil in Tibetan Buddhism, but it has to come from yourself. Nobody else can say "I'm your teacher" in this sense when you haven't connected in this way from your side, as this inspiration connection can't be forced on someone else. If any teacher says that, you know they are a false teacher, in a Buddhist tradition anyway.

Also it's an inspiration of the truth of things, and the very extensive Buddhist sutras help a lot, because if someone or something you feel is this inspiration - but then the inspiration leads you in the direction of doing things that harm others, or that is against the dharma, you know that it was a false inspiration which can happen. Especially if you are like really keen to find a guru, treat it as like a "badge" to earn or some such, you may be so desperate to find someone, that even someone who is a good teacher may become a negative inspiration for you. So this is like a safeguard and a good teacher in the Tibetan tradition will explain it to you.

I think myself that it's a shame that so many Westerners in the Tibetan tradition think they have to find a guru. I don't think many Tibetans would feel like that at all. Indeed may not even meditate, and surely most won't have gurus in this sense. Buddhists often don't meditate and are much more laid back about their religion than some of the enthusiastic Westerners.

So in this sense, if someone does have that inspiration, that his holiness the Dalai Lama is an inspiration that is connecting them to the truth of the Buddha's teachings, then yes, he is the Buddha to them. But nobody else needs to think of him as a Buddha any more than they do for anyone else who is compassionate or who has wisdom. You can still say, following the Tibetan way of thinking, that he is Avelokiteshvara, and respect him and see compassion in his actions, just as you can with anyone else who you see as compassionate in this world. It's not really that different.

There is also this idea of inspiration from a previous life that means a lot especially to Tibetans - the idea that somehow also he is connected in some intimate way with their country. But Tibet didn't always have a Dalai Lama - there was a first one, and some time there will surely be a last one. He may be the last Dalai Lama - it depends on conditions and whether it is auspicious to continue and whether they want to recognize the Dalai  Lama in his next reincarnation. 

If he chose not to do this - it's not like it would mean there still is a Dalai Lama but he is reincarnated "incognito". That's a rather cartoonish way of thinking about it.

Rather the institution of the Dalai Lama would just cease. He would be reborn in some form, but would no longer be a continuation of this inspiration stream of the Dalai Lama because the last Dalai Lama had chosen not to continue it into the future. The institution or office of the Dalai Lama is a kind of independent thing from the streams of rebirths. Something like that.

So - just a few thoughts, may help.