Who do we call Buddha? Gautama Buddha was not the only Buddha. There were many Buddhas in the past and there will be many in the future.

My question is: what characteristic of a person defines Buddha?

My answer is that Buddha is the knowledge of Truth.Anyone who has realized the Truth is the Buddha.Some say Buddha is a title ... just as King is the title. But they forget that King can be stupid i.e position of title King can be occupied by an undeserving candidate. But any person who occupies the title of Buddha must be the one who has realized the Truth.
All Buddhas of the past, present and future tell the same eternal truths again and again... they feel the same compassion and equanimity for all the beings.

My second question is : is the past Buddha an ancestor to the future Buddha? In other words can future Buddha recall the past Buddha and say that that was me in my previous manifestation ?

EDIT: I think my answer to who is Buddha is wrong one. Buddha indeed occupies an office and Brahma appoints the Buddha through a process mentioned in Manual of Buddhism by Hardy Spence. 

Any prospective Buddha needs to possess at least following three qualities:

1.Should be a Boddhisat.

2.Should have a wish to become the teacher of the three worlds.

3.Should hold the wish to release sentient beings from the evils of existence.

I quote from p.88 which translates the whole process of electing Buddha by Brahma.


  The beings who will in due course become Budhas are called Bodhisat.
  They are numberless ; but the name, in common usage, is almost
  exclusively confined to those who have become avowed candidates for
  the high office. When many ages have elapsed without the appearance of
  a Budha, there arc no beings to supply the continued diminution of the
  numbers in the brahma-lokas. This excites the attention of some
  compassionate brahma, who, when he has discovered the cause and the
  remedy, looks, out to see in what world the Bodhisat exists who will
  next become an aspirant for the Budhaship ; and when he has discovered
  the Bodhisat in question, he inspires him with the resolution that
  enables him to form the wish to become the teacher of the three
  worlds, that he may release sentient beings from the evils of
  existence. The ages that succeed this period are divided into three
  eras; in each of which we have legends of Gotama.
  1. The era of resolution (1). 2. The era of expres-, sion (2). 3. The era of nomination (3).
Dona asked this question to the Buddha in the Dona Sutta:


  "When asked, 'Are you a deva?' you answer, 'No, brahman, I am not a
  deva.' When asked, 'Are you a gandhabba?' you answer, 'No, brahman, I
  am not a gandhabba.' When asked, 'Are you a yakkha?' you answer, 'No,
  brahman, I am not a yakkha.' When asked, 'Are you a human being?' you
  answer, 'No, brahman, I am not a human being.' Then what sort of being
  are you?"
  
  "Brahman, the fermentations by which — if they were not abandoned — I
  would be a deva: Those are abandoned by me, their root destroyed, made
  like a palmyra stump, deprived of the conditions of development, not
  destined for future arising. The fermentations by which — if they were
  not abandoned — I would be a gandhabba... a yakkha... a human being:
  Those are abandoned by me, their root destroyed, made like a palmyra
  stump, deprived of the conditions of development, not destined for
  future arising.
  
  "Just like a red, blue, or white lotus — born in the water, grown in
  the water, rising up above the water — stands unsmeared by the water,
  in the same way I — born in the world, grown in the world, having
  overcome the world — live unsmeared by the world. Remember me,
  brahman, as 'awakened.'


The Buddha is the one who is awake, whereas the other unenlightened people are "asleep".

The Buddha is not an ancestor to the future Buddhas. He is able to see into future or past through some degree of omniscience, but he is not recalling them.