The Republic of Texas Again?

I doubt the secession talk is in any way serious. Secession would never survive a vote either in the state legislature, or in the electorate. I believe that it's mostly a way for right wing politicians to stroke chauvinist passions and abiding feelings of resentment which Texans of a certain age treasure like inherited jewelry.
It strikes me that most of these tax protests were in states, like Texas, that receive much more in federal funds (usually military spending, highway funding, and corporate subsidies) than they pay out. Texas was a huge beneficiary of the New Deal, especially with rural electrification and water systems. Most Texans born before 192o grew up with kerosene lamps, outhouses, and hand pumped water, as well as the constant threat of cholera, typhus, and tuberculosis.

Texas was indeed an a republic for almost 10 years; from 1836 to 1846. Yes, the annexation of Texas was very unusual, bypassing the territorial phase, with the initiative coming from the US Congress and then President Tyler. But, the annexation was put to a vote in the Texas national legislature and to the electorate. The annexation proposals won in both with lopsided majorities (I'm not sure, but I think the Tejano population, which was deeply loyal to the Republic from the beginning, may have been disenfranchised at this point; if they had been able to vote, the plurality might have been a lot less lopsided).

Texas was an accidental republic. I seriously doubt that Stephen F. Austin and the Anglo settlers who followed him had any intention of creating an independent state (or remaining Mexican citizens for that matter). I think joining the USA was always in the backs of their minds, as the Mexican government rightly suspected. Ironically, the Tejano population of Texas wanted the republic most sincerely. Texas was a center of resistance to Spanish rule long before the Anglos arrived. The Spanish general Antonio Lopez de Santa Anna, originally sent to Texas to suppress resistance to the crown, became the ruler of Mexico in 1834, nullifying the federal constitution of 1824. Anglos and Tejanos found common cause in a movement to restore the constitution and to resist Santa Anna. As the violence escalated, they soon joined in a common cause for independence with Jose Antonio Navarro, Stephen F. Austin, and Sam Houston in the leadership of this movement. When Texas became independent, few nations recognized it, most not wishing to offend Mexico. Only France, Belgium, and the Netherlands gave it official recognition, and only France under King Louis Phillipe (who had his own imperial designs on Mexico) opened an embassy in the new capital, Austin. The USA and Britain gave Texas de facto recognition (each with their own imperial designs on the territory).

I can't think of another state, except maybe Virginia, that has a bloodier history than Texas. The Texas Revolution was less a series of battles than a series of massacres and counter-massacres. In 1840, the Comanches suffered a massacre at the hands of the Republic under President Mirabeau Lamar. The Comanches retaliated with a march to the Gulf of Mexico along the Guadalupe river, massacring settlers and destroying towns as they went. Warfare with the Comanches would not end until their final surrender and exile in 1874. Piracy plagued the Gulf. There were range wars between the competing land claims of ranchers and farmers. There was labor violence. Some of the worst racial violence of the early 20th century, usually lynching campaigns and forced evictions of African Americans, happened in Texas. The only thing that was spared Texas was major Civil War battles on its territory, though Texan losses in the war were high. Texas shared with the other Confederate states the experience of defeat and military occupation. Until the New Deal, life for most Texans consisted of poverty and isolation.

PS.
Most people, including most New Yorkers, don't know that New York City considered seceding from the Union in 1861. Mayor Fernando Wood seriously entertained the idea since New York's economy was so intertwined with the Southern economy, and there was considerable sympathy with the South in the business establishment of the city.