Put simply? Probably because it replaced the office of Vice President of the Republic of Texas.

Often not well understood out-of-state is that the Republic of Texas did, indeed, function as its own country for about a decade. During this time, it had its own constitution, its own notions of presidential authority, separation of powers, and election cycles, its own bill of rights (which can actually be considered more extensive than the contemporary US!), all of which were inspired by the United States but nonetheless functioning distinctly - other revolutions in Latin America could be thought of in similar ways. Texas's constitution and politics were also uniquely defined by Mexican history, as well as the challenges it faced as a fledgling republic.

But what does this mean? Well, put simply, the Texas Constitution wasn't really written to be a state. It was adapted from the independent constitution of the Republic of Texas to make it more suited to statehood, but a lot of previous details carry forward through it. Part of that is that the office of President of Texas was subsumed into the role of Governor of Texas, and the office of Vice President of Texas was subsumed into the role of Lieutenant Governor of Texas. The Texas senate transitioned into the state senate of Texas, and you can see the pattern here. They transferred their functioning government, and particular persons were elected in the month leading up to the annexation to the US in preparation for it.

The side effect of this was that, well, the Lieutenant Governor functions more like a Vice President than a Lieutenant Governor, because the office represents just that. The Governor of Texas as an office is much weaker comparatively because it was the office of President that had checks and balances all directed against it, and the separate independent election of the Vice President (and Lieutenant Governor in turn) would count as one of those checks and balances.

I'm sorry I couldn't deliver a more thorough answer as this sub typically expects, but the answer to this question is actually fairly simple. There's not a long, drawn-out reason why this is, not some decades-or-centuries-long gradual buildup. It's just the simple transition from an independent national office to a subordinate state office while retaining most of the same powers, structure, and principles.