Learn English – How to decribe this type of relationship

Interesting question.

There really aren't any common words that express the difference between equality and equivalence. Wearing the same hat, eating the same food, driving the same car — all of these things point to equivalence rather than equality or identity.

What the hat, food and car represent here are instances of classes, but not the same instances. To express that one instance of car is identical to another instance — for example, that you and I were driving the same Ford Fusion, California License Plate No. FOOBAR1 (sorry if that is a real plate number), on the same day, I in the morning and you in the afternoon — we would have to go out of our way to express that by actually citing the plate number or explaining that I loaned you my car or you loaned me yours.

Even to say we were driving the identical car would not cause the listener, at first, to suspect we meant the exact same car with the same plate number (and serial number). Identical here would be understood only to mean we were driving the same make, model, year, and color vehicle. Even saying "the exact same" car would still be understood to mean a car exactly like the other car, not the car itself.

Look at NOAD's list of synonyms for identical:

identical adjective 1 wearing identical badges: indistinguishable, (exactly) the same, uniform, twin, duplicate, interchangeable, synonymous, undifferentiated, equivalent, homogeneous, of a piece, cut from the same cloth; alike, like, matching, like (two) peas in a pod; similar.

Not one of those synonyms expresses anything like the Law of Identity (A = A) in mathematics or the strict equality operator in some programming languages (=== instead of ==), even though the root of the word identical is, in fact, the same as for identity: Latin idem meaning "the same".

Even when we speak of things that point to identity, such as fingerprints or DNA, saying that a sample of DNA is identical to the DNA found at a crime scene does not mean the strands are the same strands, but that they come from the same person.

Yes. Continuous means from some start to some end without break:

The water flowed continuously over the dam.

whereas continual means occurring repeatedly at intervals over a time span:

I continually lose at poker.

I should also mention that continual is often substituted for continuous, and would be correct in most contexts, however the converse is not generally true. That is to say,

The water flowed continually over the dam.

is okay, but you would (hopefully) not mean

I continuously lose at poker.

since that would imply that all you do all day long is play poker and lose.