That's a good question. The modern casino emerged on a stretch of roadway south of Las Vegas in the 1940s and 1950s. What was then known as the Los Angeles Highway is now called the Las Vegas Strip.

While there have been legal casinos in Europe more or less since the Ridotto in Venice opened in 1638, Las Vegas Strip casinos set the template for modern casinos by combining the gambling floor with lodging, dining, entertainment, and retail in a way that hadn't quite been done before. It was a fusion of existing Western gambling halls with illegal casinos that were found nationwide.

When you say "relatively the same games," you probably mean slot machines and the "Big Four" table games: blackjack, craps, roulette, and baccarat. The game mix has actually evolved considerably over the past 150 years.

During the heyday of the Monte Carlo casino (the 1880s to World War I, roughly), roulette was the dominant game. Roulette itself displaced basset, which evolved into faro and which had been the game of choice in the Ridotto. Roulette was a simple but popular game that captivated Europe, but didn't quite make the transition to the United States. While illegal casinos often had a roulette wheel, faro was more often the dominant game for much of the 19th century. Known as "the game that won the West," faro was in fact a fixture in gambling halls from New York to California.

Craps was predominantly a game of the cities. It evolved from the game of Hazard somewhere in the neighborhood of New Orleans early in the 19th century, along with poker, which is a social rather than mercantile game, so a bit out of the scope of the question. For a fanciful take on the popularity of craps in the urban demimonde of the early 20th century, see Guys and Dolls and the works of Damon Runyon more broadly.

Blackjack (like roulette) is a French game that was never overwhelmingly popular, but became enough of a fixture that by the 20th century it would be offered in some gambling halls.

When Americans invented their variant of legal casino gambling in the 1930s, faro was still very much in the mix. Reno and Las Vegas gambling halls (not yet casinos) offered it. But during the 1940s, when the Strip developed, craps was the most popular game, most likely because returning GIs had played it during WWII and later the Korean War.

Blackjack became popular after Ed Thorp published Beat the Dealer in 1962. That book popularized the concept of card counting, which essentially holds that, under certain conditions, a blackjack game can become favorably to the player rather than the house. By the 1970s it had displaced craps as the top game, with roulette as the third most popular casino game. As slot machines became more interesting and offered higher jackpots, they grew to dominate as well.

Baccarat is a relatively recent arrival. Though the game itself is older than craps for certain, it was never very popular in the United States. The Sands introduced baccarat to Las Vegas as a novelty in the late 1950s, and it remained a game with a bit of a mystique and high minimum bets.

Across the world in Macau, however, baccarat emerged to become the dominant table game in the casinos of Stanley Ho from the 1960s onward. It became the favored game of Chinese high rollers, and it began to surge in Las Vegas with the influx of Chinese high rollers starting around 2005. Here's a report showing the evolution of table mix on the Las Vegas Strip.

Due to the increasing globalization and standardization of gambling, those four games, along with slot machines, now dominate. And due to regulation, the rules tend to be similar and the games tend to be dealt in the same way regardless of jurisdiction (with some notable exceptions).

Anyway, that's a quick answer.

Source: my very own Roll the Bones: The History of Gambling (Casino Edition).