I'm going to write a fresh answer because the response I wrote elsewhere in this thread was focusing on a major error. Best to start afresh and just answer the main question: 'what would Jesus have considered the year to be?'
The answer is that in the Roman province of Syria during Jesus' time the standard practice was to refer to years by the regnal year of the Roman emperor.
New year in the Syrian calendar corresponded to 1 October in the Julian calendar, so when Tiberius took the emperorship in August 14 CE, that was the start of 'Tiberius 1'. Two months later, in October, began 'Tiberius 2'.
So for example the most popular ancient choice for the date Jesus died, spring 29 CE, would have been 'Tiberius 16', which began in October 28 CE.
You can in fact see this system being used at one point in the New Testament: Luke 3.1-3 puts Jesus' baptism and the start of his ministry in 'the fifteenth year of Tiberius Caesar's reign'. As a backup it also lists a number of local contemporary rulers and a magistrate.
As in my other post, I recommend E. J. Bickerman's Chronology of the ancient world (2nd ed. 1980) as a guide to these matters. I do not recommend Wikipedia, or pretty much any other encyclopaedia, as they can be guaranteed to contain lots of misinformation about ancient chronography.
In addition to the answers r/Holy_Shit_HeckHounds linked to I want to mention that different people kept time in different ways. Today most of humanity has agreed in one standard, but that wasn't the case back then
The Romans used to count years since the foundation of Rome using the Julian Calendar. This is the system the Catholic Church later adapted and became the standard we use today
Jewish people at the time of Jesus were beginning to use what we know today as the Hebrew Calendar, which starts counting form the year that was thought to be the creation of the world which in our calendar is 3761 BC. Jewish people still use this calendar to this day, mostly for their religious celebrations, but it's important to mention that at the time of Jesus they used a primitive version of it and it's likely only the educated elite cared about it
Other civilizations have used other strategies to tell time.
For example Egyptians used the Egyptian Calendar, which is similar to the Roman Calendar, probably because the Romans copied it form them. edit: u/KiwiHellenist has pointed out why this statement is a misconception at best, thanks. However while this calendar was useful to keep time inside a year to keep track of the years themselves they used the reign of pharaohs. For example they would say "in the day of the moon of the eighth year of the reign of Tutmosis" or things like that. One instance of this are the letters of Heqanakht (I hope it's fine to link to a video I have about them), this is how he marks the date when he's writing the letter
In fact most ancient civilizations kept track of time with the reign of kings and emperors. We have tablets written by the Assyrians, the Babylonians, the Hittites and many other Middle East civilizations which are just lists of kings and the time they reigned. These lists have been used to know when events in the history of these nations happened. For example the Egyptians mention a solar eclipse happening during the rule of a certain pharaoh, which we can use to know when it happened. Then if we know in what year Ramses II of Egypt reigned, and we know he fought a war against Hattusilli III of Hatti, and we have the list of kings of Hatti then we can know when every king of Hatti reigned. Then if Hattusilly fought a war against the Assyrians for example then we can know when the Assyrian kings reigned... it's a sort of chain of decipherment
However it's never that simple. In the case of Egypt there are 2 solar eclipses in the span of 10 years, which one is the eclipsed mentioned in the records?, no one has any idea, but hey, an uncertainty of ten years is not that bad. Also often the list of kings have mistakes, or they include mythical kings, or maybe a new king had deposed the previous one and wanted to erase them from history so there are inconsistencies in the lists... but it can all be sorted out
Long story short. Most civilizations had calendars to keep track of time within a year and a few of them used to count years from one important event but must of them just used their rulers
An old answer from the FAQ by a deleted user: BC / AD - what did they use at the time? by [deleted]
Also potentially of interest Did the Romans calculate the year different when the Julian calendar was created, and if yes, when did it change into the present days year representation. written by u/Astrogator (scroll down to their second reply for the answer; their top level is a clarification)
Also an answer on a thread from a year and a half ago by u/toldinstone: