Clearly Noam Elkies' answer is optimal for CM curves. For non-CM curves that are a quadratic twist of one another, the probability is $1/2$, because the probability that $a_p=0$ is $0$ by Chebotarev's density theorem.

For a non-CM and CM curve, or two non-CM curves that are not a twist of each other, let $G$ be the image of the Galois group in $GL_2(\mathbb Q_\ell) \times GL_2(\mathbb Q_\ell)$ acting on the Tate modules in both curve. The subset of $G$ consisting of elements that have the same trace in both representations is closed in the Zariski topology of positive codimension, hence has $\ell$-adic measure $0$, by Chebotarev's density theorem a zero-density set of primes live there.

Hence Noam Elkies' example is optimal, and we can take any $\delta> 3/4$ to imply equality up to isogeny.

In general, the Sato-Tate conjecture is true for sets which are closed in the Zariski topology, because those sets are also closed, and hence measurable, in the $\ell$-adic topology, and we know equidistribution for the $\ell$-adic measure by Sato-Tate.

The $7/8$ bound is attained if instead of representations of elliptic curves we allow two-dimensional Artin representations. Take an extension of Galois group $D_4 \times \mathbb Z/2$, and compare the two-dimensional irreducible representation of $D_4$ to its quadratic twist by $\mathbb Z/2$.

Edit: There is a purely algebraic proof of the $7/8$ upper bound. In general:

Let $V$ and $W$ be two distinct $n$-dimensional irreducible representations of a compact Lie group $G$. The probability that $tr(V)=tr(W)$ is at most $1-1/2n^2$.

Proof: Let $X$ be the locus where $tr(V)=tr(W)$ and let $Y$ be the locus where it is not. Let $p$ be the measure of $Y$. Then by Schur's lemma, the definition of $X$, Schur's lemma again, and the trivial estimate $|tr(V)| \leq n$, $|tr(W)| \leq N$:

$0 = \int_G tr(V) \overline{tr(W)} = \int_X tr(V) \overline{tr(W)} + \int_Y tr(V) \overline{tr(W)} = \int_X tr(V) \overline{tr(V)}+ \int_Y tr(V) \overline{tr(W)} = 1- \int_Y tr(V) \overline{tr(V)} + \int_Y tr(V) \overline {tr(W)} \geq 1 - 2 p n^2$