Do other state Constitutions have Unique clauses like that of Texas?

With regard to splitting itself into new states, the answer to this is both "all of them" and "none of them, including Texas." Territories coming into the Union have had special conditions placed on their statehood, most famously those requiring them to exclude slavery (Ohio, Indiana, Illinois, Michigan, Wisconsin, Iowa, Minnesota, and West Virginia) but I haven't read of any state getting special privileges. Slavery restrictions generally went into state constitutions, though they could in potential be revised out by amendments or a new constitution thereafter. Illinois had a serious effort to do just that early in its history as a state.

The claim that Texas has this power hails from the joint resolution annexing it to the United States:

Third -- New States of convenient size not exceeding four in number, in addition to said State of Texas and having sufficient population, may, hereafter by the consent of said State, be formed out of the territory thereof, which shall be entitled to admission under the provisions of the Federal Constitution; and such states as may be formed out of the territory lying south of thirty-six degrees thirty minutes north latitude, commonly known as the Missouri Compromise Line, shall be admitted into the Union, with or without slavery, as the people of each State, asking admission shall desire; and in such State or States as shall be formed out of said territory, north of said Missouri Compromise Line, slavery, or involuntary servitude (except for crime) shall be prohibited.

That doesn't quite say that Texas has the unilateral power to declare itself split into as many as five states. It rather declares that Texas might see division in the future and lays out some conditions for it. This is because the Texas Republic claimed a staggering amount of territory, much larger than the present state. Texas exerted actual control over rather less land. The most famous case of that is the land claimed south of the Nueces River, the border dispute that contributed to keeping Texas out of the United States when it first asked for annexation. Less famously, Texas' did not agree on its western border with the New Mexico territory. Texas wanted the Rio Grande River as its boundary, which would give half of modern New Mexico to it. One usually sees that boundary on maps of the situation.

During the crisis of 1850 this came close to blows. The dispute had a strong sectional dimension, since Texas definitely had slavery but New Mexico might potentially turn into a free state (and by Mexican law, already was free soil) down the road. Texas prepared to enforce its claim by force of arms. That would probably mean attacking the US Army in New Mexico. Many southerners rallied, at least rhetorically, to the Texan cause. President Zachary Taylor, himself a southerner, stood firm against them. It might have come to blows, but Taylor ate some tainted refreshments at a Fourth of July celebration, got sick, and died a month later. Eventually a more moderate solution prevailed. Texas remained gigantic, and a bit more gigantic than antislavery Americans had hoped, but not quite so gigantic as it had dreamed. In exchange, it got Congress to assume responsibility for the Lone Star Republic's debts.

That's some background, but what did the provision grant? Texas remained massive and largely unsettled by white Americans. In situations like that in the past, portions of previously organized territories would be admitted as states and the remainder consolidated into a new territory. Thus the Northwest Territory had turned into Ohio, with the rump becoming the Indiana Territory, etc. California did this for itself, laying out borders deliberately clear of anything Texas aspired to to grease the wheels of its own admission to the Union, but I don't know if its practice was typical or not. Texas skipped the territorial stage, with some controversy, which left it with that hinterland that might see future subdivision in different circumstances. The lines in the resolution recognize that.

However, any such division would require Texas' consent. It was a state, after all. States, per the Constitution, cannot be divided without their approval:

no new states shall be formed or erected within the jurisdiction of any other state; nor any state be formed by the junction of two or more states, or parts of states, without the consent of the legislatures of the states concerned as well as of the Congress.

So Texas has the same veto power as any other state. However, it doesn't really possess the ability to unilaterally subdivide itself into states. Congress must consent and can't legally be forced to accept that land as states. The Constitution says new states "may be" admitted but not that they must be. (Congress has, however, accepted faits accompli on this front.) The wording is "shall be entitled to admission", just like another other territory, rather than something like "become at once" members of the Union. This was not an entirely theoretical issue New York had previously consented to the admission of Vermont, Virginia to Kentucky, and Massachusetts to Maine.

Thus we have Texas with the power to split itself into up to four new states (and a rump Texas), practically speaking, only with the consent of Congress. The spinoffs could then apply for admission to the Union. But that power isn't really unique, as any other state could petition Congress to do the same and would have to get the same approval. Likewise their territorial integrity is protected from Congressional alteration unless they consent to it. There were occasional proposals during the late Antebellum that Texas should divide into new slave states to help restore sectional parity in the Senate, often with the idea that doing so would help defuse the growing sectional strife over slavery. A deadlocked Senate wouldn't do much to enrage proslavery southerners and would serve very well to frustrate the aims of the northern and sometimes antislavery majority in the House.

Technically, Congress laid a further restriction on Texas with the resolution. By the letter of the law, with its consent any other state could be split into any number of new states. Texas could have no more than four new neighbors but Missouri or wherever could be split into an arbitrary number. That's really unlikely to ever have happened, of course. The size of Texas is what made splitting it seem viable. Any of Texas' states that came in north of the Missouri Compromise line would also have to exclude slavery, but that just extended a restriction that also applied across the Louisiana Purchase until 1854, when the Kansas-Nebraska Act overthrew the system amid tremendous controversy.

TL;DR? Texas doesn't really have the power to split itself at will. Congress must still agree. It does have the power to do so if Congress gives the OK, but that's the same power every other state has. So it has a power, but not quite the one usually claimed, and it's not a unique power to Texas.

Sources, aside materials quoted: William W. Freehling's Road to Disunion, Volume One James McPherson's Battle Cry of Freedom Allan Nevins' Ordeal of the Union, Volume One David Potter's The Impending Crisis