There are lists of known finite 2-transitive groups, and for examples other than those of affine type (i.e. those with a regular normal elementary abelian subgroup), it should not be hard to show that there are no more examples.
But I would expect there to be examples of affine type, and by searching through the Magma database of primitive permutation groups I found an example of degree 343. These are
PrimitiveGroup(343,49) and PrimitiveGroup(343,50) in Magma. Irritatingly, the numbering is different in GAP, and (hoping I have got this right), they are numbers 67 and 68 in the GAP list.
The two groups in question have order $343 \times 342$ - so they are sharply 2-transitive. Their stabilizers are isomorphic nonabelian groups of order 342, with centres of order 6.
I would conjecture that there are infinitely many examples of a similar type.
Added: The group ${\rm A}\Gamma{\rm L}(1,7^3)$ has structure $7^3:(7^3-1):3$, and has four subgroups of index 3. The examples here are two of those subgroups. One of the other two is ${\rm AGL}(1,7^3)$, which is also sharply 2-transitive, but has cyclic point stabilizer. The fourth does not act 2-transitively. There are corresponding subgroups of ${\rm A}\Gamma{\rm L}(1,p^3)$ for all prime $p \equiv 1 \bmod 3$, so there are indeed infinitely many examples. More generally, for $p,q$ primes with $p \equiv 1 \bmod q$, there will be $q-1$ non-isomorphic sharply 2-transitive groups of degree $p^q$ with isomorphic stabilizers.