Bukhari

Hadith Commentary:
That is, on the day when they were accustomed to fasting, for example, Monday or Thursday, and if on that day the day of Ashura also occurred, they would fast. The day of Ashura refers to the tenth date of Muharram al-Haram; this has been a historical day since ancient times.

Hadith Commentary:
The fast of Ashura was initially obligatory.
After the obligation of Ramadan was established, the obligation of Ashura was abrogated; however, its recommendation still remains.
In the narration from Aisha (radi Allahu anha), this is further clarified. She said:
The Quraysh used to fast on the day of Ashura during the pre-Islamic period of ignorance.
The Messenger of Allah (sallallahu alayhi wa sallam) himself also fasted on Ashura while residing in Makkah and made his companions observe this fast as well.
When the fasts of Ramadan were made obligatory, he (sallallahu alayhi wa sallam) said:
“Whoever wishes may fast on Ashura, and whoever wishes may leave it.”
(Sahih al-Bukhari, al-Sawm, Hadith: 1893)

Hadith Commentary: Benefits and Issues: By "the Days of Allah" are meant those days in which Allah, the Exalted, bestowed favors and blessings upon the previous prophets and their communities. That is why, in Surah Ibrahim, it is stated:
﴿وَذَكِّرْهُمْ بِأَيَّامِ اللَّهِ﴾ "And remind them of the Days of Allah," that is, admonish and advise them regarding the favors and blessings of Allah, the Exalted.
And (remind them) that He destroyed and annihilated their enemies.

Benefits and Issues:
All days indeed belong to Allah, but those days in which a particular event has occurred and which hold significance from a religious and legal (shar‘i) perspective are referred to as (Ayyām Allāh) — “the Days of Allah.”

Benefits and Issues:
➊
From this, it is understood that this fast is not obligatory; however, it is certainly a deed of reward.

➋
Any practice from the Age of Ignorance (Jahiliyyah) that is endorsed by the Qur’an and Hadith becomes a ruling of our Shari’ah. Thereafter, it is not performed considering it an act of Jahiliyyah, but rather as a command of Islam. And any practice that has been prohibited becomes absolutely forbidden (haram). As for any matter regarding which there is no evidence of command or prohibition, it is better to abstain from it, because the Prophet (sallallahu alayhi wa sallam) opposed the Jews and Christians in many matters. To such an extent that the noble Companions (radi Allahu anhum) understood that opposing the disbelievers is a principle of Islam. This is the reason that when there was consultation regarding announcing the time of prayer, the Companions (radi Allahu anhum) rejected the suggestions of ringing a bell or lighting a fire, as these were the ways of non-Muslims.

For details, see: (Sunan Ibn Majah, al-Adhan, Chapter: The Beginning of the Adhan, Hadith: 707)