Let $\kappa=\beth_\kappa$ and $\mathcal L_{\kappa\kappa}$ be the infinitary logic allowing $<\kappa$ conjunctions and $<\kappa$ quantifications. If $\mathfrak A$ is a $v$-structure with $|A|<\kappa$ and $|v|<|A|$. Can we state that $|Th_{\kappa\kappa}(\mathfrak A)|=|A|<\kappa$?
Where $Th_{\kappa\kappa}(\mathfrak A) = \{\varphi\in\mathcal L_{\kappa\kappa}:\varphi$ is a sentence and $\mathfrak A\models\varphi\}$.
If I understand what you're asking (per your comments), the answer is yes via a general construction.
Suppose $v$ is a signature and $\mathfrak{A}$ is a $v$-structure. There is an infinitary sentence characterizing $\mathfrak{A}$ up to isomorphism amongst $v$-structures; roughly, it has the form $$\exists (x_i)_{i<\mu}[\forall y(\bigvee_{i<\mu}y=x_i)\wedge(\bigwedge_{i<j<\mu}x_i\not=x_j)\wedge \mathbb{A}]$$ where $\mu=\vert\mathfrak{A}\vert$ and $\mathbb{A}$ is basically the atomic diagram of $\mathfrak{A}$ pulled back along a fixed bijection $\{x_i: i\in\mu\}\rightarrow\mathfrak{A}$. This is exactly analogous to how we show that every finite structure in a finite language is pinned down up to isomorphism by a single first-order sentence.
Now let $\theta=\max\{\mu, \vert v\vert, \omega\}$; the sentence above belongs to $\mathcal{L}_{\theta^+,\mu^+}$, so a fortiori to $\mathcal{L}_{\theta^+,\theta^+}$. In particular, if $\kappa>\theta$ then the $\mathcal{L}_{\kappa,\kappa}$-theory of $\mathfrak{A}$ is determined by a single $\mathcal{L}_{\kappa,\kappa}$-sentence.