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No doubt the Prophet ﷺ  said “I was sent to perfect good character”, which is one of the key factors of peaceful coexistence in our social environment and relationships, in any given human society. However, the hadith does not mean that he was only sent for that purpose, but that this purpose of good morals and behavior fulfills the greater purpose of Dawah i.e. spread of Islam and invitation for all towards the path of guidance, the way of Allah ﷻ. Especially when doing Dawah (invitation) towards Islam, one should always be mindful of this essential factor because it makes the mission much more effective and inspiring. The best example of this given in the Quran is when Allah ﷻ commanded Prophet Musa (alaihi salaam) to visit and invite Pharaoh towards guidance and Islam:

(Al Quran 20:43–44) Go, both of you, to Pharaoh, for he has exceeded all bounds. Speak to him gently so that he may take heed, or show respect.

Despite the fact that Allah ﷻ knew Pharaoh had “exceeded all bounds”, He still advised Musa (alaihi salaam) and his brother Haroon (alaihi salaam) to be at their best in behavior and character when delivering His Message because our behavior impacts what we are delivering. Since Prophet Muhammad ﷺ knew this very well, his emphasis on improving human behavior was the greatest, keeping in mind the fact that after his passing away, Muslims bear the responsibility to deliver the message in its correct form to those who are ignorant i.e. Non-Muslims as well as Muslims.

Original answer written on Dec 30, 2019 - Is fasting on 10th of Muharram, the day of Ashura a fabrication in Islam? 


Prophet Muhammad ﷺ ordered the Muslims to fast this day after he saw the Jews fasting this day as a celebration because Allah ﷻ saved them from Pharaoh on this day. Hence he said: “I am closer to Moses than they (Jews)." So, he observed the fast (on that day) and ordered the Muslims to fast on it.” (Sahih al-Bukhari 3397) 

Unfortunately some circles among Shia Muslims question the credibility of this hadith of fasting on Ashura because they allege that Prophet’s beloved grandson Imam Hussain (alaihi salaam) was killed on this day hence the Bannu Umayyad Caliphs shifted the focus of this crime by introducing this hadith among the Muslims so that it is considered a day of celebration. To support this claim they give the following reasons for questioning its credibility:

1. They say that the hadith is narrated by four companions of the Prophet: Ibn Abbas (r.a), Abu Musa Al-Ashari (r.a), Abu Huraira (r.a), Muawiya, and all four were either too little to remember (Ibn Abbas was 4 yr old) or not present in Madinah till after the Battle of Khaybar which took place after 7th year of Hijra (emigration).
2. They also say that the term “Ashura” surfaced after Imam Hussain was killed on the tenth of Muharram. After that, Ashura came to be known as the tenth of Muharram, but before that it was simply the tenth day of any month.
3. That there is no fasting for Jews on the day God saved Moses till today and in the past also they never fasted for this occasion. They only fast on Yom Kippur when Moses returned from Mt. Sinai.
4. They question the wording of the hadith: "When the Prophet (ﷺ) arrived at Medina, he noticed that some people among the Jews used to respect Ashura." , and claim that Prophet entered Madinah in Rabi-ul-Awwal month and not in Muharram. Hence it must be a forgery.
5. Prophet is following the Jews, why would he do it when he is a Prophet of Allah ﷻ, why would he need to follow the Shariah of Jews?
   
   

Answer to the First point

Firstly the hadith narration is not only limited to four companions, in fact it has also narrated by an Ansari (Helper) companion named Abu Qatada Al Ansari (r.a) who was an older companion when the Prophet ﷺ  was in Madinah and the hadith is found in Sahih Muslim:
Abu Qatada al-Ansari (Allah be pleased with him) reported that the Messenger of Allah (ﷺ) was asked about his fasting: “He was asked about fasting on the day of 'Ashura (10th of Muharram), whereupon he said: It expiates the sins of the preceding year.” (Sahih Muslim 1162 b)

Additionally, it was also narrated by the most intimate and beloved wife of the Prophet, Ayesha (r.a) who knew all his teachings and practices. As for the four companions Ibn Abbas (r.a), Abu Musa Al-Ashari (r.a), Abu Huraira (r.a) and Muawiya may not have been in Madinah or were very young yet that does not nullify the fact that they were aware of what the practices of the Prophet and his companions in Madinah used to be. They were not very little at the time of narrating this incident, they were reputable scholars in Madinah, Kufa and Makkah where they were teaching traditions to other Muslims. Also the fasting of Ashura was made optional (nafl) once Ramadaan became obligatory (fard) in Madinah but before Ramadaan became obligatory (fard) the Ashura fast was of obligatory nature. Hence all Muslims in Madinah strongly adhered to this act of worship. Such as Ibn Umar (r.a) said: “But when (the order of compulsory fasting) in the month of Ramadan was revealed, the Prophet said, "It is up to one to fast on it (i.e. day of 'Ashura') or not." (Sahih Al Bukhari 4501)

Answer to the second point 

As for the term “Ashura” it did exist even in the Jewish tradition. The word "tenth" in Hebrew is Asarah or Asharah (עשרה), which is from the same Semitic root A-SH-R. For instance the in the Torah Book of Leviticus designates the tenth day of the seventh month as holy and a fast (Lev. 16, Lev. 23, Num. 29) and the Jewish term used for the 10th day is Asharah (עשרה). Leviticus 16 / Hebrew. Most importantly the Quraysh used to observe fast in Makkah on this day even in pre-Islamic times as Narrated Aisha (r.a) : “'Ashura' (i.e. the tenth of Muharram) was a day on which the tribe of Quraish used to fast in the prelslamic period of ignorance.” (Sahih Al Bukhari - 3831) 

Answer to the third point
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The modern Jews don’t fast on this day which is correct. We also need to keep in mind that Jews, all of them do not follow same thought, there are various sects and groups that have existed in the past and still exist. Some have survived down to our times while others perished in history. For instance the Quran claims that Jews worship Uzair (Ezra) as the son of God but modern Jews don’t follow that practice but the Jews of Madinah did follow this belief and hence they were a separate group that carried that belief: (Al Quran 9:30) AND THE JEWS say, "Ezra is God's son," while the Christians say, "The Christ is God's son." Such are the sayings which they utter with their mouths, following in spirit assertions made in earlier times by people who denied the truth!" [They deserve the imprecation:] "May God destroy them!" - Whats more interesting to note is that the day of Ashura usually coincides with the 10th day of Tishri one of the months of Hebrew Calendar. Tishri usually falls in September - October of the global Gregorian Calendar and in the year of 2018 both Ashura and tenth of Tishri fell on the exact same day. This year (2019) the 10th Ashura was on September 10th and 10th Tishri was on September 11th. The tenth day of Tishri is the Jewish holiday of Yom Kippur which is still a day when Jews fast! Even though Yom Kippur is celebrated because Prophet Musa (alaihi salaam) came back from Mt. Sinai. Nevertheless it does show some coincidence.

Answer to the fourth point

Prophet Muhammad ﷺ was already fasting himself before he moved to Madinah. After he moved to Madinah he ordered the Muslims to fast with him and made it obligatory (fard) for everyone as narrated Aisha (r.a): “The Prophet (ﷺ) also used to fast on this day. So when he migrated to Medina, he fasted on it and ordered (the Muslims) to fast on it.” (Sahih Al Bukhari 3831) This pre-Islamic tradition may have been because of the holiness of this day in the history of mankind such as the incidents of:

• Allah created the heavens, mountains, oceans, pen (qalam) and tablet (lawh) on this day.
• Hazrath Ibrahim (alaihi salaam) was born, and Allah ﷻ saved him from the furnace on this day.
• Pharaoh was drowned,
• Allah ﷻ forgave the transgression Adam (alaihi salaam) ,
• Isa (alaihi salaam) was born on this day
• The day of Reckoning will occur on this day.
(Al Ghunya Talibi Tariq al-Haqq )

The apparent wording of some ahadith claim that “he started fasting when he came to Madinah” doesn’t mean that his arrival happened in Muharram. It may have happened in some other month but the order to fasting was given to Muslims in the month of Muharram after his arrival in Madinah and thats what it means.

Answer to the fifth point

Prophet Muhammad ﷺ was not only following Jews in their fasting on the day of Ashura. In fact he followed also their Qiblah (Jerusalem) before Kabah (Makkah) was finalized as the Qiblah for Muslims. He also gave Shariah punishments in according to the Jewish law such as stoning as well as others practices. These he followed to prove that his revelation is not separate from what was taught by the previous prophets and their followers. Nevertheless he ordered the Muslims to fast an extra day with the 10th of Muharram (i.e. 9th or 11th), so that in this act Muslims are considered distinct from Jews and Judaism.

References:

https://sunnah.com/bukhari/60/70 
https://sunnah.com/muslim/13/253
https://sunnah.com/urn/41830
https://sunnah.com/bukhari/63/57
https://www.mechon-mamre.org/p/pt/pt0316.htm 
https://sunnah.com/bukhari/63/57
http://sunnah.org/wp/2008/07/18/historical-significance-ashura/