$DLO$, the first order theory of dense linear ordering without endpoints, is $\omega$-categorical. So once you prove that the Fraisse limit, say $M_{0},$ of all finite linear ordering is a model of $DLO$, you already have proved that $M_{0}$ is isomorphic to$\langle\mathbb{Q},<\rangle,$ since $DLO$ is the same as $Th(\langle\mathbb{Q},<\rangle).$ This can be done using the universality and the homogeneity of the Fraisse limit.

The fact that $DLO=Th(\langle\mathbb{Q},<\rangle)$ is obtained using Vaught's test and the facts that $DLO$ has no finite models and is categorical in an infinite cardinal.
The argument for the $\omega$-categoricity of $DLO$ is a back-and-forth argument on two countable models that is interesting for its part.

**Also, to see more directly** why the Fraissé limit of finite linear orders is the ordering of the rationals rather than that of the integers, note that the only difference between these orders is that the rationals are dense, i.e. for any two distinct elements of $\mathbb{Q}$ there is another element laying in between. To show that the order of that Fraissé limit is the ordering of the rationals you need to use the *universality* or *richness* of a Fraissé limit.

Supose that $F$ is the Fraissé limit of the class $\langle\mathcal{K},\subseteq\rangle$ in a language $\mathcal{L}$. $F$ is *universal* or *rich* in the following sense:

For any finite $A\subseteq F$ and any $B\in\mathcal{K}$ containing $A$ as a substructure, there is an $\mathcal{L}$-embedding (order-preserving in the case of linear orders) of $B$ into $F$ that extends the identity map over $A$.

Now, for given distinct elements in $F,$ say $a_{1}$ and $a_{2}$. w.l.g. we may assume that $a_{1}<a_{2}$ (we presumed we have proved that $F$ is linearly ordered). By taking $A:=\{a_{1},a_{2}\}$, there is a 3-element linear order containing $A$ with an extra element $b$ that lives in the class $\mathcal{K}$ and have the following order

$a_{1}<b<a_{2}.$

Now, using richness of $F$, there exists such element inside $F$. This proves that the limit $F$ is dense.