I found myself wondering the same thing a couple of weeks ago. Even with the restriction that $p > 2$ and the residual representation is irreducible, it does not follow that $E_{1}$ and $E_{2}$ have the same primes of bad reduction, nor that $a_{\ell}(E_{1}) \equiv a_{\ell}(E_{2}) \pmod{p}$ for primes of bad reduction.

For example, take $p = 5$ and $E_{1} : y^{2} + xy + y = x^{3} + 4x - 6$ (aka $X_{0}(14)$). The mod $5$ Galois representation attached to $E_{1}$ is surjective. For any elliptic curve $E$, Rubin and Silverberg write down an isomorphism $X_{E}(5) \cong \mathbb{P}^{1}$, the modular curve parametrizing elliptic curves $F$ so that $F[5]$ and $E[5]$ are isomorphic via and isomorphism respecting the Weil pairing. For $E = E_{1}$, one can take $E_{2} : y^{2} + xy = x^{3} - x^{2} - 4492x + 126416$. This curve has $E_{1}[5] \cong E_{2}[5]$, but $E_{2}$ has bad reduction at $5$ and $E_{1}$ has good reduction at $5$. It is not surprising that one can fail to have $a_{\ell}(E_{1}) \equiv a_{\ell}(E_{2}) \pmod{p}$ when one of $E_{1}$ and $E_{2}$ has good reduction at $\ell$ and the other doesn't.

One can also have $a_{\ell}(E_{1}) \not\equiv a_{\ell}(E_{2}) \pmod{p}$ when both curves have bad reduction at $\ell$. Looking in the same family (of curves directly $5$-congruent to $E_{1}$) one can find curves $E_{2}$ and $E_{3}$ that have multiplicative reduction at $199$ where one curve has $a_{199}(E_{2}) = -1$ and the other has $a_{199}(E_{3}) = 1$. The examples I found are a bit horrendous:
$$ E_{2} : y^2 + xy = x^3 - x^2 -
183192520591565859491828595856323163822228663229886627562x +
1062123179152064591727007023640066384957014639073044998393946451273297951607053468340$$
and
$$ E_{3} : y^2 + xy + y = x^3 -
187739104428369548177010660620161617529156118721103898180850571x +
1108087782673654453656602976287807709340970590450649179565293267500311360376488672021879300982. $$