It is an immediate consequence of Cohen's forcing that if there is one countable transitive model of $\sf ZFC$, then there are many of them. Even if all of these models are of the same height, there are still many.

But the proof of existence of generic filters don't carry over for uncountable models of set theory. Not without further assumptions (like Martin's axiom) anyway.

Is it possible to have an uncountable transitive model which is the only uncountable transitive model? Since that model is going to have to be $L_\alpha$ for some $\alpha>\omega_1$, is it also consistent with $V\neq L^1$, that a unique uncountable model exists?

Can this be pushed arbitrarily high (perhaps with, or without the assistance of $\sf MA$ and the Tennenbaum-Solovay theorem) so there are many transitive models of size $<\kappa$, but only one of size $\kappa$?

Footnotes:

1. Clearly one can just add a Cohen real to the universe without damaging any uncountable sets in a mind-shattering way. And on the other hand, if $V\neq L$ in a strong enough way, i.e. $0^\#$ existing in the universe, then there are arbitrarily large uncountable transitive models of $\sf ZFC$. Simply $L_\kappa$ for $\kappa$ a cardinal, or a Silver indiscernible.

So my purposely vague question about $V\neq L$ is some middle-ground between the very uninteresting $L[r]$ for some "simple" real number $r$, and the existence of large cardinals (which automatically implies there are many uncountable transitive models).

It is an immediate consequence of Cohen's forcing that if there is one countable transitive model of $\sf ZFC$, then there are many of them. Even if all of these models are of the same height, there are still many.

But the proof of existence of generic filters don't carry over for uncountable models of set theory. Not without further assumptions (like Martin's axiom) anyway.

Is it possible to have an uncountable transitive model which is the only uncountable transitive model? Since that model is going to have to be $L_\alpha$ for some $\alpha>\omega_1$, is it also consistent with $V\neq L^1$, that a unique uncountable model exists?

Can this be pushed arbitrarily high (perhaps with, or without the assistance of $\sf MA$ and the Tennenbaum-Solovay theorem) so there are many transitive models of size $<\kappa$, but only one of size $\kappa$?

Footnotes:

1. Clearly one can just add a Cohen real to the universe without damaging any uncountable sets in a mind-shattering way. And on the other hand, if $V\neq L$ in a strong enough way, i.e. $0^\#$ existing in the universe, then there are arbitrarily large uncountable transitive models of $\sf ZFC$. Simply $L_\kappa$ for $\kappa$ a cardinal, or a Silver indiscernible.

   So my purposely vague question about $V\neq L$ is some middle-ground between the very uninteresting $L[r]$ for some "simple" real number $r$, and the existence of large cardinals (which automatically implies there are many uncountable transitive models).