What is the largest prime number $p$ for which one knows examples of nonisogenous elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$ with isomorphic mod $p$ Galois representations: $E_1[p] \cong E_2[p]$? I do not require the isomorphism to preserve the Weil pairing.

The largest known is for $p = 17$. This can be found by searching Cremona's tables. In particular, in this paper, Tom Fisher mentions the examples of curves 3675b1 and 47775b1, which are

$E_{1} : y^{2} + xy + y = x^{3} + x^{2} - 393x - 9654$

and

$E_{2} : y^{2} + xy + y = x^{3} + x^{2} + 398095697x + 4077685984826$.

The conjecture that if $p > 17$, there are no such curves is known as the Frey-Mazur conjecture. (Frey and Mazur originally asked whether this can occur for $p > 5$, because in that situation the curve $X_{E}(p)$, the moduli space of elliptic curves with the same mod $p$ Galois representation as $E$, has genus $> 1$.)