I am assuming following meanings for equality and equivalence
- equality: exact sameness (as philosophical identity)
- equivalence: belonging to the same class
Neither of these definitions that I chose are complete and you can find other, different meaning for these words, especially in technical sense (logic, philosophy, mathematics, science). However, I choose these as I think these are close to what you meant and my answer is based on them.
Regarding these differences, let me mention for example, in mathematics equality is a relation that defines equivalence class (any relation that is reflexive, symmetric and transitive i.e. partitions a set so that every element of the set is a member of one and only one cell of the partition, is equivalence relation).
In logic the definition of equivalence is very different (see material equivalence and logical equivalence), and I think it does not apply here, though some usage of the word can imply logical equivalence.
Regarding the equality in the sense of philosophical identity or exact sameness, usually in English that is easily expressed. Your first example can be shortened further
It was the same man.
and it still expresses this1.
The definite article and adjective "same" applied to non-abstract words express the exact sameness2. If you substitute "man" with "car" or even seemingly indistinguishable "ant" you will express the exact sameness.
You can try to extends it to abstract concepts
It was the same proof.
However, abstract concepts are abstracted and the exact sameness holds for the concept, not for the instance. Instances can indeed be different and, at the same time, equivalent. But, the concept of the proof remains one (and only, hence exactly same to itself).
Now, for the equivalence - let me here rewrite your second example in the same structure
It was the same dish that her cousin had ordered.
what I propose is that here with the term "a dish that her cousin had ordered" the exact sameness remains. "A dish" here is not the food on the plate, but "food prepared and cooked in a particular way", 2nd meaning of the noun.
In this case, context and frequency determine that we are not talking about the 1a meaning: "the food contained in a dish".
A more ambiguous example would be
It was the same car that her cousin had ordered.
In this case car can indeed refer to an instance of a car or a type of the car (model, color, etc) and in this case it is really not possible to distinguish what is meant.
In such cases what FumbleFingers and teylyn mention does help
It was the very same car that her cousin had ordered.
Also here there are other helping phrases: the exact same car, the very same car, literally the same car, actually the same car. All these try to emphasize that the meaning of "the same" is more true than without them. However, they also work in constructions where you want to emphasize that it is the same model (especially if such context had been implied, for example by talking about very rare, discontinued model).
Now, I believe you can be more specific by dropping "the same" and say:
It was the very car that her cousin had ordered.
where without the relation "the same" you can only interpret it as an actual material instance of an object (and avoid hyperbolic interpretations 3). Terms actual and exact can be interchanged here with the same meaning (also "It was literally the car...").
1 However, do note that the context remains, it is only unspecified now. We still compare one idea with another: the man is the same as the man we saw previously, or the man in photograph, or the man in your dream, etc.
2 Definitive article comes from demonstratives meaning that.
3 For example "literally" is so often used figuratively (as hyperbole) that the dictionary entires list the second meaning which is opposite to the original meaning! See here.