Although you have tagged this with “John O’Sullivan” and “Annexation,” I do not believe this question really has to do with that article except in a tangential way. The article does not really talk about what Texas was like in 1845. Instead, it simply asserts that Texas is more closely tied to the US than to Mexico. Therefore, I will answer by looking at evidence from outside O’Sullivan’s article. I will look at the population of Texas, its political system, and its laws with respect to slavery. Texas resembled the US in all of these ways.
In 1845, Texas was dominated by Anglo-Americans. Even though it had once been part of Mexico, it was not predominantly Hispanic. We can see this from the identities of the men who ran the government of the Republic of Texas. To be sure, a Hispanic man, Lorenzo de Zavala, helped to write the constitution and was elected interim vice president by the delegates to the convention that wrote the constitution. However, once the offices of president and vice president were put to a popular vote, every single candidate for either office was a white man. This was similar to the United States, where white men dominated politics and where the bulk of the population was white.
In 1845, Texas allowed slavery. Slavery was against the law in Mexico, but not in the United States. This was one of the causes of the Texas Revolution. The Americans who moved to Texas when it was part of Mexico did not want to give up their slaves. In this way too, the Republic of Texas was like the United States.
Finally, the Republic of Texas was very much like the United States in its political system. It had a democratic political system. It had orderly and free elections (which Mexico lacked at that time). It had a bicameral Congress and separation of powers between three branches of government. Many parts of the constitution were taken directly from the Constitution of the US.
In these ways, Texas was very much like the United States in 1845, even before it had actually been annexed as an official part of the US.