Traditionally, the Hebrew calendar was based on a lunar calendar, with each month starting with the sighting of the moon (source). Months were either 29 or 30 days. This is exactly the same as the Hijri calendar, and the calendar observed in the Arabian Peninsula prior to Islam.
The Hebrew calendar continued to be lunar until the fourth century when Hillel II (head of Sanhedrin at the time) changed it from lunar to fixed based on mathematical and astronomical calculations. This was done to coordinate between a lunar year and a solar year. Since then, the Hebrew calendar has 12 or 13 months of 29 or 30 days using a 19-year cycle with leap years occurring in years 3, 6, 8, 11, 14, 17 and 19 (source).
With this change, the Islamic and Jewish calendars would only partially match. For example, the beginning of the 1438 Hijri year and the Hebrew 5777 year were one day apart (2nd and 3rd October, respectively). Next year, the new year will start on 21 September foth both calendars (mathematically), and so on, until the 5780 leap year when the Hebrew calendar will be reset.
As mentioned in this hadith in Sahih Muslim
Ibn 'Abbas reported that the Messenger of Allah arrived in Medina and found the Jews observing fast on the day of 'Ashura. The Messenger of Allah said to them: What is the (significance) of this day that you observe fast on it? They said: It is the day of great (significance) when Allah delivered Moses and his people, and drowned the Pharaoh and his people, and Moses observed fast out of gratitude and we also observe it. Upon this the Messenger of Allah said: We have more right, and we have a closer connection with Moses than you have; so Allah's Messenger observed fast (on the day of 'Ashura), and gave orders that it should be observed.
Pesach (Passover) refers to God passing over the houses of the Jews when slaying the firstborn of Egypt (source). This is mentioned in Exodus 12:12-13:
For I will go through the land of Egypt in that night, and will smite all the first-born in the land of Egypt, both man and beast; and against all the gods of Egypt I will execute judgments: I am the LORD. And the blood shall be to you for a token upon the houses where ye are; and when I see the blood, I will pass over you, and there shall no plague be upon you to destroy you, when I smite the land of Egypt.
It is not related to the day the Pharaoh and his people drowned, as per the hadith above.
Rosh Hashanah, Yom Kippur, and Sukkot
‘Ashura is most likely Yom Kippur, which is the 10th of Tishri.
Rosh Hashanah means "head of the year" and it occurs on the first and second days of Tishri, and it is known as the Jewish New Year (source), being the Sabbatical and Jubilee new year out of multiple new years observed in the Hebrew calendar (Nissan 1 for calendar counting reign of kings, Elul 1 for the tithing of animals, etc.). So Rosh Hashanah is the one that matches onset of Muharram in the Hijri calendar.
Sukkot, on 15th of Tishri, the fifth day after Yom Kippur, commemorates the onset of the 40-year wandering period for Bani Israel (source), which happened right after the drowning of the Pharaoh. As mentioned in Numbers 33:3-5
And they journeyed from Rameses in the first month, on the fifteenth day of the first month; on the morrow after the passover the children of Israel went out with a high hand in the sight of all the Egyptians, while the Egyptians were burying them that the LORD had smitten among them, even all their first-born; upon their gods also the LORD executed judgments. And the children of Israel journeyed from Rameses, and pitched in Succoth.
Exodus 33:3-5 shows that (1) the first month of the year is Tishri, and (2) Sukkot is on the 15th of Tishri, five days after Yom Kippur. I am not sure how to reconcile that Passover is 15th of Nissan (Exodus 12 and 13, and Leviticus 22) and Exodus 33 with Passover being on 14th of Tishri.
Yom Kippur, 10th of Tishri, is a day that Jews refrain from work, fast and attend synagogue services. It is the Day of Atonement that traditionally is the first day after the Bani Israel’s exodus from Egypt and arrival at Mount Sinai (source). This also ties Yom Kippur to the drowning of the Pharaoh.