Did people from New York City's 5 boroughs originally consider themselves to be part of the same city?

They did not. They were separate cities through much of the 19th century. City agglomeration became a tend in the 19th century, as advances in public transit (e.g. steam ships and street cards) helped tie metro areas together and create new suburbs and increased municipal services (fire to departments, water, etc) gave incentives for towns to consolidate. Read, for example, Walt Whitman’s poem “Crossing Brooklyn Ferry”—the poem was written when Brooklyn and New York (Manhattan) were separate entities. Where Brooklyn-native Whitman’s ferry crossed is apparently today where the Brooklyn Bridge is. (I’m going to guess you live in New York—the bomb ice cream shop Ample Hills takes their name from a line in the poem.) To give you a sense of how different Whitman’s Manhattan was, he wrote the poem in 1855, two years before Central Park was established. The first consolidated institution, the Metropolitan Fire District, between Brooklyn and New York came not long after in 1865. Both were still decades before consolidation.

Today we know the Twin Cities as Minneapolis and St. Paul, but for much of the 19th century this was used to refer to other cities, including New York and Brooklyn. The first reference to them as the twin cities in the NYT is from 1859, but refers to Pittsburgh and Allegheny City (Allegheny became a part of Pittsburgh in 1907), comparing them to New York and Brooklyn. The first use of twin cities in the NYT to refer to explicitly refer to New York and Brooklyn in the Times comes only slightly later, in 1860:

For instance, Mr. CONKLING presented a memorial from New-York, asking for the passage of a law creating a Health Department for the Cities of New-York and Brooklyn, (a Metropolitan dodging of the Constitution,) and that provision be made for the formation and enforcement of a sanitary system, proved to be so necessary by the increased bills of mortality […] I wonder not that they are solicitous for the health of the good people of the twin cities. It is in the public health -- and the consequent ability to pay -- that these gentlemen find their reward.

In an 1872 article called “A Tale of Three Cities”, the Times compares mortality rates for New York, Philadelphia, and Brooklyn (I believe during a particularly severe heatwave). Again, though, we see the fairly common attribution of “the twin cities of New-York and Brooklyn” (there was als an 1870 article in the Times about elections that search says mentions “twin cites” but I couldn’t find. Interestingly, it broke down election results for New York, for Brooklyn, then “Returns from Cities and Towns”, which covered the rest of the state. For whatever reason, there’s no returns from Queens County or Richmond County reported, but you’ll notice there are returns for Kings County that aren’t listed with Brooklyn. Why?

At that time, Brooklyn did not cover all of Kings County and there were several other towns in the county. Curbed actually has a really fun little history of Brooklyn and Kings County, but have you ever heard someone refer to “South Brooklyn” and mean like Cobbel Hill, Park Slope, Red Hook areas and not the parts of Brooklyn that are actually furthest south? That’s because in the 19th century, before consolidation (the consolidation of all of Kings County into Brooklyn was a process separate from but related to the consolidation of all of the Five Boroughs into New York City), this was the southern limit of Brooklyn. Williamsburg(h) and Bushwick were also separate municipalities. Where was that border? Not surprisingly, at Division Ave. There were several splits in the 19th century (for example, New Lots breaking off from Flatbush and Williamsburg(h) breaking off from Bushwick), peaking in the period 1852-1854: the cities of Brooklyn and Williamsburg(h), and the towns of Bushwick, New Lots, Flatlands, Gravesend, New Ultrecht, and Flatbush.

In 1854, Brooklyn started swallowing up its smaller neighbors, Bushwick and Williamsburg(h) first. They then tried unsuccessfully to annex the rest of the county in 1873, but successfully annexed New Lots in 1886, everything but Flatlands in 1894, and finally Flatlands 1896. But, other than Williamsburg and Bushwick, these areas were small and still largely rural. In the big 1894 annexation, for instance, Brooklyn had a population of nearly a million. The biggest town they annexed was Flatbush, which had a population of just 12,625.

By the time New York and Brooklyn (and the rest of the Five Boroughs) consolidated, Brooklyn was either the fourth largest city in the county, behind only New York, Chicago, and Philadelphia, but much much larger than Boston and St. Louis, which only had about half a million residents each. This is actually a slight step down for Brooklyn due to Chicago’s rapid growth. In the 1860, 1870, and 1880 censuses, Brooklyn was the third largest city (in the 1840 and 1850 censuses, before the Bushwick consolidation, it was the 7th largest city). Fun fact: before Philadelphia consolidated in 1854 (before that, Philadelphia was just today’s Center City), two of its future neighborhoods (Northern Liberties in 1840, Spring Garden in 1850) are among the ten largest cities in the country.

New York was actually consolidated in 1898, which was before the subway. The main mass transit was still ferries (which may help somewhat explain why Staten Island was included, /u/aamirislam) and street cars. But look at a city that didn’t consolidate as aggressively, Boston. The MBTA serves all the “inner ring” suburbs (once known as “street car suburbs”) like Cambridge, Brookline, and Somerville. The first streetcars, interestingly, were horse-drawn rather than electric or internal combustion and, in the Boston area, actually connecting Boston to Cambridge from the very start. Here, I think the difference from San Francisco and New York were more bridges than municipal boundaries: there were long bridges across the Charles, but the East River was harder to navigate, never mind San Fransisco Bay. The first bridge between Brooklyn and Manhattan was of course the Brooklyn Bridge, which was started in 1869 and only finished in 1883. The next bridge to tie Manhattan to what are now the outer boroughs (the Williamsburg Bridge) wasn’t started until shortly before consolidation (1896) and wasn’t finished until after consolidation (1903). The first bridge across the San Francisco Bay didn’t open until 1937. By that point, there were six bridges connecting Manhattan to Brooklyn and Queens and I think maybe eight connecting Manhattan to the Bronx, with another bridge set to open in 1939 connecting the Bronx and Queens. The biggest period of subway growth was in the early 20th century. After 1950’s, there was less of an emphasis on rail travel and more of an emphasis on cars. Americans often take New York’s subway as typical, but it is really quite extraordinary, not just compared to other American cities, but compared to the rest of the world. For a long period of time, it was the only (or at least one of the very few) subway which ran 24 hours a day.

So, getting back to the original question, while of course the residents of what became the outer boroughs knew they were in the New York area, they didn’t always want to be part of New York (or even Brooklyn). When Brooklyn tried to annex the rest of Kings County in 1873, a local official from New Ultrech was quoted in the Times as saying:

The government of New-Utrecht, he said, was the least expensive; it was only an agricultural village, a long way from Brooklyn, and not in need of being included in the expenses of municipal government, at least not for the present.

The soon to be consolidated towns of Kings County were small and, very often, still dominated by old Dutch farming families. Brooklyn often had similar cultural reasons for their reticence to join with Manhattan: Brooklyn was largely Protestant, whereas Manhattan had many more recent immigrants like Jews, Italians, and the Irish. These fears were particularly pronounced mid-century, closer to things like the 1863 Draft Riots, but they lingered on. The title of one 1894 NYT article: “Greater New York in Doubt: the City Votes For It but Brooklyn is Uncertain” (then, in smaller type, the City of Churches’ [Brooklyn] Figures Held Back, It Is Said, By Politicians Who Want To Keep Their Power, Long Island City Gives a Majority of 2376, and All Richmond County [Staten Island] Is Willing to Join the City). To give a sense of how small the other boroughs were at the time, while New York was 1.8 million in 1890, and Brooklyn over 950k, Long Island City (the largest town in what would become Queens) was 30.5k, Flushing was 19.8k, Jamacaia was 14.4k, etc. Its worth noting that in this same period, much of Westchester was considered for consolidation, but ultimately decided against it.

While other areas seem to want access to both services and the future growth potential (making their real estate more valuable), as far as I can tell one major reason the already developed city of Brooklyn finally decided to join New York was water: New York City had already built an extensive system to bring in water from upstate, but Brooklyn still relied on local aquifers on Long Island.