Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Yes, and your reasoning is more or less correct.

Given a field $F$, $F[\alpha] = F(\alpha)$ if and only if $F[\alpha]$ is a field if and only if $\alpha$ is algebraic over $F$.

Note that if $\alpha$ is transcendental, then $F[\alpha]$ is isomorphic to the polynomial ring $F[x]$, which is not a field.

Now if $F$ is a field and $\alpha_1$ and $\alpha_2$ are both algebraic over $F$, $$F[\alpha_1,\alpha_2]\cong F[\alpha_1][\alpha_2] \cong F(\alpha_1)[\alpha_2]\cong F(\alpha_1)(\alpha_2)\cong F(\alpha_1,\alpha_2),$$ since $\alpha_2$ is also algebraic over $F(\alpha_1)$.

If $F$ is a field and $\alpha$ is algebraic over $F$, then $F[\alpha] = F(\alpha)$ always.

Explanation: $c_n \alpha^n + \cdots + c_0 = 0$ so $-\frac{1}{c_0}\left(c_n \alpha^{n-1} + \cdots + c_1\right) \cdot \alpha = 1$, so $\alpha$ is invertible in $F[\alpha]$. (If $c_0 = 0$ then factor some power of $\alpha$ out of the equation.) Similar logic finds an inverse for any expression involving a linear combination of $\alpha$ and its powers (i.e. the general element of $F[\alpha]$).

But if $\alpha$ is transcendental, then $F[\alpha] \cong F[t]$ is not a field, and is a strict subring of $F(\alpha) \cong F(t)$.

More generally, if $A$ is a domain which is a finite-dimensional algebra over a field $F$, then $A$ is a field.

Indeed, take $a \in A$, $a\ne 0$, and consider $\phi: A \to A$ given by $\phi(x)=ax$. Then $\phi$ an $F$-linear transformation. Moreover, $\phi$ is injective because $A$ is a domain. Since $A$ is finite-dimensional algebra over $F$, $\phi$ injective implies $\phi$ surjective, and so $1$ is in the image of $\phi$ and $a$ has an inverse in $A$. This inverse can be found by expressing $\phi$ with respect to a basis of $A$ and solving a linear system.

$\mathbb{Q}[\sqrt{2},\sqrt{3}]$ is clearly a domain because it is contained in $\mathbb{R}$.

$\mathbb{Q}[\sqrt{2},\sqrt{3}]$ is also a finite-dimensional algebra over $\mathbb{Q}$. Indeed, $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ is the set of all polynomial expressions in $\sqrt{2}$ and $\sqrt{3}$ with coefficients in $\mathbb{Q}$, and so a basis is $\{ 1, \sqrt{2}, \sqrt{3}, \sqrt{6} \}$ because all powers greater than $1$ can be reduced.

Another way is to notice that $\mathbb Q(\sqrt2, \sqrt3)=\mathbb Q(\sqrt2+\sqrt3),$ and $(\sqrt2+\sqrt3)^{-1}=\sqrt3-\sqrt2\in\mathbb Q(\sqrt2, \sqrt3)=\mathbb Q(\sqrt2+\sqrt3),$ so $$\mathbb Q(\sqrt2, \sqrt3)=\mathbb Q(\sqrt2+\sqrt3)=\mathbb Q[\sqrt2+\sqrt3].$$
Now it follows easily that $\mathbb Q(\sqrt2,\sqrt3)=\mathbb Q[\sqrt2,\sqrt3].$ In fact, this works for every finite separable extension, mutatis mutandis.
Hope this helps.