A few corrections first:
I have heard that when the prophet met the people of the book, he asked them why they were fasting in the day of Asura. They replied that it was a good day.
This meeting was not with people of the book in general but with the Jews of Medina once he settled there! Most scholars say this was not on the day he came to Medina but on the first encounter of 'Ashura' there so this conversation has taken place in the year 2 A.H..
See for example in sahih al-Bukahri with three different chains ending with Sa'id ibn Jubayr from Ibn 'Abbas and two with Shoa'bah from Abu Bishr from the former quoted:
- Abu Ma'mar -> Abdul-Warith -> Ayyub -> Abdullah ibn Sa'id ibn Jubair -> His father (Sa'id ibn Jubayr) -> Ibn 'Abbas
- Mohammad ibn Bashar -> Ghundar -> Shoa'bah -> Abu Bishr -> ...
- Ya'qub ibn Ibrahim -> Rooh -> Shoa'bah -> Abu Bishr -> ...
However there's a version of this hadith quoting the Christians beside the Jews too you can find it in sahih Muslim and with a similar narrator chain (... from Yahya ibn Ayyub from Ismael ibn Ummayyah (al-Qurashi) from abu Ghatafan from ibn 'Abbas... is a common part!), in sunan abi Dawood. As I'm not aware of a Christian practice on this topic I assume this addition is false or comes from lack of knowledge and wrong assumptions, or maybe at the time Christians still used to celebrate this happening. But Allah knows best!
The prophet then said he would fast that day and he added that he would observe two fasts in the next Asura (in the next year) if he is alive.
The Prophet (pbuh) used to fast Ashur'a as the Mekkan pagans used to fast this day (it was apparently the day the covers of the Kaaba were changed... according the Arabic wikipedia see here and here) even before. As a statement of ibn Jarir says:
كانت الكعبة فيما مضى إنما تكسى يوم عاشوراء، إذا ذهب آخر الحجاج حتى كان بنو هاشم، فكانوا يعلقون عليها القمص يوم التروية من الديباج، لأن يرى الناس ذلك عليها بهاء وجمالاً، فإذا كان يوم عاشوراء علقوا عليها الإزار
The Kaaba in the past was newly covered (with the Kiswah) on the day of 'Ashura', when the last pilgrims left until Banu Hashem were responsible then they used to hang on it veils (made of
silk brocade) on the day of at-Tawriyah (8th of Dhil-Hijjah, at this day people used to drink and gather water from Zamzam before leaving Mekka towards Mina), so that people (visitors) see the beauty and majesty of the Kaaba, then when the day of 'Ashura' came they hanged the usual veil on it!
So he asked the Jews (of Medina) about their reason to fast, ahadith are showing that he and (as it seems) the muhajireen (Quraish) were fasting that day.
See for example: in sahih al-Bukhari (sane chain), al-Bukhari again (with an other chain: the teacher of al-Bukahri and his teacher), in a-Shamaa'il al-Mohammadya (three new narrators), Jami' at-Tirmidhi (same as in a-Sahmaa'il, but he said it was also narrated on the authorithy of ibn Mas'ud, ibn 'omar, Qais ibn Sa'd, Jabir ibn Samura and Mo'awiya), sahih Muslim (Muslims teacher and his teacher are new in the chain) and al-Muwatta' the source of al-Bukhari's first version).
After knowing that it was the day Allah saved Musa (pbuh) and bani Israel from Pharao he ordered this fast but as fasting Ramadan was ordered in the year 2 A.H. this was the single occasion were fasting 'Ashura' was an ordered fast, that's why believers who were not fasting were asked not to eat in hadiths such as this or this in sahih Muslim!
The statement that he would observe fasting the 9th and 10th came much later as it is a statement the Prophet (pbuh) made, but couldn't keep as he died before reaching the next 'Ashura' so you must distinguish between these statements. I'd like to add that even shi'a sources recommend this fast!
Here the sunni sources: Sahih Muslim on the authorithy of ibn 'Abbas
As you see your Question is basically a mix of different narrations most of them have been shortly explained in my answer here Why do some Muslims fast on the 9th and 10th moharram even when it was abandoned by the Prohpet (saw)?.
I'd like to add that all these ahadith are quoted at least in one of both sahihs, so the chance of it having weak narrators are very few.
Now to your Questions:
Did he say that?
I think I made clear what he (pbuh) said or not in my "correction".
Did he say that he would observe two fasts in the next Asura?
As stated above he said this but died before this next 'Ashura'. A reason for declaring to fast those two days was to differ from the fast of the people of the book, as (at least) Jews were fasting the 10th only as all narration agree.
Is it a weak hadith that says this or something similar?
As stated above all these narrations and many more can be found in the sahih books verifying the authenticity of this amount would take weeks or months for a layman as I'm. I found in this site a takhrij (a detailed source identification quoting all the sources and chains) of all the ahadith quoting the fast of 'Ashura' (unfortunately it is in Arabic) and all of the details I've quoted here are quoted there too (If I would have known this site earlier I would have spared a lot of time ;)), so there you may even find a lot of ahadith I missed in my posts. There's also a nice article (in Arabic too) quoting the historical development of the fast of 'Ashura' with good explanations.
Why the Prophet followed the Jews in fasting that day?
Here some responses:
Al-Maaziri (may Allah have mercy on him) said concerning this issue and the response to it:
What the Jews say is not to be accepted (and taken as true), so it may be that the Prophet (blessings and peace of Allah be upon him) received revelation that confirmed what they said, or that he heard about that from many different sources, so that he concluded that it was true. End quote.
This was quoted by an-Nawawi in Sharh Muslim, 8/11
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Even though this fast was not originally intended to do the same as the people of the Book, his saying, “We are closer to Moosa than you” is a confirmation of the prescription to observe this the fast, and explaining to the Jews that what you do of expressing love for Moosa, we do too, and we are closer to Moosa than you.
Iqtida’ as-Siraat al-Mustaqeem, p. 174
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said:
Whatever the case, he (blessings and peace of Allah be upon him) did not fast it in emulation of them – i.e., the Jews – because he used to fast it before that, and that was at the time when the Prophet (blessings and peace of Allah be upon him) liked to do the same as the people of the Book in matters that he had not been forbidden to do so.
Fath al-Baari, 4/248
Also note that the Prophet (pbuh) liked to follow the people of the scripture unless he had a clear revelation on a matter as stated in ahadith such as
Narrated 'Abdullah bin `Abbas:
The Prophet (ﷺ) used to keep his hair falling loose while the pagans used to part their hair, and the People of the Scriptures used to keep their hair falling loose, and the Prophet (ﷺ) liked to follow the People of the Scriptures in matters about which he had not been instructed differently, but later on the Prophet (ﷺ) started parting his hair. (Sahih al-Bukhari, you'll find it in other hadith collections too.)
Note if you go one hadith back -168 instead of 169- in sahih al-Bukhari you'll find on of the narrations quoting our Prophet's (pbuh) saying "We are closer to Moses than you."!
And if he heard of a victory of the Romans (ahl-al-Kitab) he was happy for that as stated in this hadith and here on the authority of a different narrator.
For further details I also recommend to read the following two fatwas from islamqa on 'Ashura' and previous religions and on the virtue of fasting it.