Suppose we take the set of real numbers $\mathbb R$ as undefined objects and assume the existence of two binary operations defined on $\mathbb R$, denoted by $+$ and $\cdot$ and called 'addition' and 'multiplication', respectively, i.e., $+:\mathbb R\times\mathbb R\rightarrow\mathbb R$ and $\cdot :\mathbb R\times\mathbb R\rightarrow\mathbb R$, such that the field axioms are satisfied.
Suppose we also assume that $\mathbb R$ is equipped with an order relation denoted by $\lt$ and called 'less than' such that $\mathbb R$ is an ordered field.
Of course, we have that, for instance,
(A1) $\forall a,b,c\in\mathbb R:(a+b)+c=a+(b+c)$.
But the notion of 'equals' has not been defined nor explicitly introduced as an undefined concept. Isn't it necessary to state something like, 'we assume the existence of a binary relation = on $\mathbb R$, such that the reflexive, symmetric, and transitive properties are satisfied'?
Lastly, I've used the terms 'binary operation' and 'relation'. I have a working definition of these terms, i.e., a binary operation on a set $S$ is a function from $S\times S$ to $S$ and a relation on a set $S$ is a subset of $S\times S$. As far as I know, the only undefined concepts I am using are $\mathbb R$ and 'equals' ($=$), but I'm not sure if this is correct.
Ultimately the answer depends on what meta-logic you are using. Let me start with the boring answers and move to more interesting ones.
Since your definition of binary operation implies $\mathbb{R}$ is a "set", then, if you are using a typical set theory, you already have a global notion of equality.
Maybe your set theory has a bunch of ur-elements and $\mathbb{R}$ is a set of those. Then you would need to specify what equality meant for those ur-elements, but this would be part of articulating your meta-logic, i.e. your set theory, not part of defining $\mathbb{R}$.
For an exercise like this, it's much more typical to take a logical approach where we define a logical theory whose models will be the objects in which we are interested. This approach is much closer to the exercise you seem to be performing. In this case, $\mathbb{R}$ is not a set but a sort. The question, though, is still "what is the logical framework within which you are working?"
The traditional answer is first-order logic (FOL). FOL does have a global notion of equality, so again, you need not specify it. On the other hand, you will need to specify enough constraints so that your models are actually the reals. It turns out this is not possible in first-order logic and this is the basis for some approaches to non-standard analysis. Using second-order logic or higher order logic (HOL) is an option, e.g. Tarski's approach, but this may not be a problem for you. In fact, it may be a benefit.
There are other logics that don't bake equality in such as FOLDS and various forms of type theory. In such logical frameworks you will indeed have to specify that some relation exists that is an equivalence relation and a congruence.
The theories of doctrines and institutions provide a means to abstract over the particular choice of logical framework.