What makes this tricky is that metric or continuous logic (in the sense of this document) doesn't directly allow for quantification over $\varepsilon$'s and $\delta$'s like that. You can still express it with a schema, but what's difficult in this case is that you have quantification over elements of the structure before quantifying over $\varepsilon$ and $\delta$. There's also some subtlety with how you formalize implication in this context.

Generally speaking a statement is only going to be preserved by ultrapowers/metric elementary equivalence if it's *uniformly* true in the structure. This fact is baked into the formalism in some places, such as the requirement to give a modulus of uniform continuity for a given function. If a function in a metric structure fails to be uniformly continuous, then in some elementary extension it won't even be a function. More generally purely 'topological' properties are often too fragile to be preserved under ultrapowers unless they are actually true in some uniform way.

EDIT2: This is not as straightforward as I thought. ~~I can tell you roughly when this is not going to work. If there is an $\varepsilon > 0$ such that for every $\gamma > 0 $ there exists vectors $x,y$ with $\| x\| = \| y\|=1$ such that the $\delta$ needed to satisfy your condition is $<\gamma$, then in an ultrapower (over a countable index set) the condition will fail. So I would guess that in order for the condition to be satisfied in ultrapowers of the structure you actually need the stronger form you mentioned where the $(\forall x)(\forall y)$ quantifiers are after the $(\exists \delta > 0)$ quantifier. And I mean 'need' in a strong sense, i.e. if you have a Banach algebra such that your condition is true in all ultrapowers then the stronger form of the condition (although restricted to norm 1 vectors) actually holds in the first place.~~

EDIT: ~~I think that maybe you can construct a counterexample from the proof of theorem 4.12 here, specifically some kind of product algebra of the algebras $\ell_1(\mathbb{Z}_{n!})$ with the convolution algebras.~~ EDIT3: This reference is still relevant but I was being too optimistic about openness of multiplication being preserved under products.