Is every sigma-algebra of cardinality $2^\kappa$ for some cardinal $\kappa$?
Not necessarily.
Suppose that $2^{\aleph_0} = \aleph_1$ and $2^{\aleph_1}=\aleph_3$. It is still the case that $\aleph_2^{\aleph_0} = \aleph_2$.
Now take a collection of $\aleph_2$ subsets of $\omega_1$ and let them generate a $\sigma$-algebra. It will have size $\aleph_2$ which is not a cardinal of any power set.
I think the following counterexample works in ZFC.
Set $\lambda = \beth_{\omega_1}$, so that $\lambda = \bigcup_{\alpha < \omega_1} \beth_\alpha$. Consider the $\sigma$-algebra $\mathcal{F}$ consisting of all countable and co-countable subsets of $\lambda$. ("Countable" here will mean "finite or countably infinite").
I claim $|\mathcal{F}| = \lambda$. It is clear that $|\mathcal{F}| \ge \lambda$ because $\mathcal{F}$ contains all the singletons.
For the other direction, let $\mathcal{F}_0$ consist of all the countable subsets of $\lambda$, and $\mathcal{F}_1$ all the co-countable subsets. Suppose $A \subset \lambda$ is countable, so $A = \{x_1, x_2, \dots\}$ (where I allow the sequence to be finite). For each $n$, there exists $\alpha_n < \omega_1$ with $x_n \in \beth_{\alpha_n}$. Letting $\alpha = \sup_n \alpha_n < \omega_1$ (since there are either finitely or countably many $n$), we have $A \subset \beth_\alpha$; that is, $A \in \mathcal{P}(\beth_{\alpha})$. So, $\mathcal{F}_0 \subset \bigcup_{\alpha < \omega_1} \mathcal{P}(\beth_\alpha)$. But $\mathcal{P}(\beth_\alpha)$ by definition has cardinality $\beth_{\alpha + 1} < \lambda$. Thus, $\mathcal{F}$ is contained in a union of $\omega_1$ sets each having cardinality less than $\lambda$, so $|\mathcal{F}_0| \le \omega_1 \cdot \lambda = \lambda$ (since $\lambda > \omega_1$). There is clearly a bijection between $\mathcal{F}_0$ and $\mathcal{F}_1$, so $\mathcal{F} \le \lambda + \lambda = \lambda$.
On the other hand, $\lambda$ is not of the form $2^\kappa$. If $\kappa \ge \lambda$ then $2^\kappa > \lambda$ by Cantor's theorem. And if $\kappa < \lambda$ then $\kappa \le \beth_\alpha$ for some $\alpha < \omega_1$, so that $2^\kappa \le 2^{\beth_\alpha} = \beth_{\alpha + 1} < \lambda$. In either case, $2^\kappa \ne \lambda$.
Moreover, $\beth_{\omega_1}$ is consistently the least possible cardinality of a counterexample.
Proposition. Assume GCH. Then every $\sigma$-algebra $\mathcal{F}$ with $|\mathcal{F}| < \beth_{\omega_1}$ has $|\mathcal{F}| = 2^{\kappa}$ for some $\kappa$.
Proof. We already know it's true for $\mathcal{F}$ finite, so suppose $\mathcal{F}$ is infinite. By GCH, every cardinal is a beth number, so $|\mathcal{F}| = \beth_\alpha$ for some $\alpha < \omega_1$. Since we know that there are no countable $\sigma$-algebras (this can also be seen by a variant of the argument below), we have $\alpha > 0$. I claim that $\alpha$ must be a successor ordinal; then we are done, for if $\alpha = \beta+1$ then $|\mathcal{F}| = \beth_{\beta + 1} = 2^{\beth_\beta}$. So suppose $\alpha$ is a limit ordinal; it is countable, so $|\mathcal{F}| = \beth_\alpha$ has countable cofinality, so we can write $\mathcal{F} = \bigcup_{n=1}^\infty \mathcal{C}_n$ where $|\mathcal{C}_n| < |\mathcal{F}|$ for all $n$. Suppose without loss of generality that every $\mathcal{C}_n$ is uncountable and that $\mathcal{C}_1 \subset \mathcal{C}_2 \subset \dots$. Now let $\mathcal{F}_n = \sigma(\mathcal{C}_n) \subset \mathcal{F}$. As noted by Joel David Hamkins on MathOverflow, we have $$|\mathcal{F}_n| = |\mathcal{C}_n|^{\aleph_0} \le 2^{|\mathcal{C}_n|} < |\mathcal{F}|.$$ The first inequality holds by identifying a function $f : \aleph_0 \to \mathcal{C}_n$ with a subset of $\aleph_0 \times \mathbb{C}$, where $|\aleph_0 \times \mathcal{C}_n| = |\mathcal{C}_n|$ because $\mathcal{C}_n$ is infinite. The second inequality is because $\mathcal{F}$ is a limit beth number.
Hence we have $\mathcal{F} = \bigcup_{n=1}^\infty \mathcal{F}_n$, where $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dots$ and each is properly contained in $\mathcal{F}$. Passing to a subsequence, we can suppose they increase strictly. But as shown in [1], a countable strictly increasing union of $\sigma$-algebras can never be a $\sigma$-algebra, so this is a contradiction.
[1] Broughton, Allen; Huff, Barthel W., A comment on unions of sigmafields, Am. Math. Mon. 84, 553-554 (1977). ZBL0372.60004. Thanks to hot_queen for this reference.
More generally, this can easily be extended to show that for every limit ordinal $\alpha$ of uncountable cofinality, there exists a $\sigma$-algebra of cardinality $\beth_\alpha$ which is not of the form $2^\kappa$ for any $\kappa$; and under GCH, no other cardinality is possible.