The usage of anonymous enums

That's a so-called enum trick for declaring a compile-time integer constant. It's advantage is it guarantees that no variable is instantiated and therefore there's no runtime overhead. Most compilers introduce no overhead with integer constants anyway.

If this is old code, then enum might have been used for the "enum hack".

You can learn more about the "enum hack", for example, in this link: enum hack

Enums don't take up any space and are immutable.

If you used const int color = 1; then you would solve the mutability issue but if someone took the address of color (const int* p = &color;) then space for it would have to be allocated. This may not be a big deal but unless you explicitly want people to be able to take the address of color you might as well prevent it.

Also when declaring a constant field in a class then it will have to be static const ^{(not true for modern C++)} and not all compilers support inline initialization of static const members.

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Disclaimer: This answer should not be taken as advice to use enum for all numeric constants. You should do what you (or your co-workers) think is more readable. The answer just lists some reasons one might prefer to use an enum.