No, it doesn’t mean that Texas has rights under the Constitution that other states don’t have. The admission of any new states formed from Texas must be admitted according to the Constitution. There is no doubt about that.
There was/is no treaty. There was a joint resolution and order of annexation, neither of which trumps or conflicts with the Constitution.
Just for info I’ll publish the Joint Resolution so those that care to can read it in its entirety.
Joint Resolution for Annexing Texas to the United States
Approved March 1, 1845
Resolved by the Senate and House of Representatives of the United States of America in Congress assembled,
That Congress doth consent that the territory properly included within and rightfully belonging to the
Republic of Texas, may be erected into a new State to be called the State of Texas, with a republican form
of government adopted by the people of said Republic, by deputies in convention assembled, with the consent
of the existing Government in order that the same may by admitted as one of the States of this Union.
2. And be it further resolved, That the foregoing consent of Congress is given upon the following conditions,
to wit: First, said state to be formed, subject to the adjustment by this government of all questions of
boundary that may arise with other government, —and the Constitution thereof, with the proper evidence of
its adoption by the people of said Republic of Texas, shall be transmitted to the President of the United States,
to be laid before Congress for its final action on, or before the first day of January, one thousand eight
hundred and forty-six. Second, said state when admitted into the Union, after ceding to the United States all
public edifices, fortifications, barracks, ports and harbors, navy and navy yards, docks, magazines and
armaments, and all other means pertaining to the public defense, belonging to the said Republic of Texas,
shall retain funds, debts, taxes and dues of every kind which may belong to, or be due and owing to the said
Republic; and shall also retain all the vacant and unappropriated lands lying within its limits, to be
applied to the payment of the debts and liabilities of said Republic of Texas, and the residue of said lands,
after discharging said debts and liabilities, to be disposed of as said State may direct; but in no
event are said debts and liabilities to become a charge upon the Government of 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.
3. And be it further resolved, That if the President of the United States shall in his judgment and discretion
deem it most advisable, instead of proceeding to submit the foregoing resolution of the Republic of Texas, as an
overture on the part of the United States for admission, to negotiate with the Republic; then,
Be it resolved, That a State, to be formed out of the present Republic of Texas, with suitable extent and
boundaries, and with two representatives in Congress, until the next appointment of representation, shall be
admitted into the Union, by virtue of this act, on an equal footing with the existing States, as soon as the terms and conditions of such admission, and the cession of the
remaining Texian territory to the United States shall be agreed upon by the governments of Texas and the United
States: And that the sum of one hundred thousand dollars be, and the same is hereby, appropriated to defray the
expenses of missions and negotiations, to agree upon the terms of said admission and cession, either by treaty
to be submitted to the Senate, or by articles to be submitted to the two houses of Congress, as the President may direct.
Approved, March 1, 1845.
https://www.tsl.state.tx.us/ref/abouttx/annexation/march1845.html