How to compare units?

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Something is confusing me, it's about real world units vs abstract ones and what should be abstract and absolute.

Here's my problem:

1 dog + 1 dog = 2 dogs

A dog is an abstract unit, all dogs are different, yet this equation still makes sense. You can't convert a dog to a number of equal particles (so you could relate it to other objects) and say for example:

1 dog = 5u
2 dogs = 5u + 5u = 10u

In the other hand, if you have a dog named Morgan (a unique dog), you could say:

1 Morgan = 3u
1 Morgan + 1 Morgan = 2 Morgans = 6u

But if you do this:

1 Morgan + 1 dog = 2 dogs
1 dog = 1 Morgan = 3u

How do I say that Morgan is a dog and relate them both?

Does it make sense to have any absolute units? Because we don't know everything and everything might change in different places or times.

Thank you.

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Well I find something wrong with this equation:

1 dog = 1 Morgan = 3u

Morgan is a dog, but a dog is not Morgan. It's like saying because a human is a mammal, a mammal is a human. But, Morgan is more specific than just "dog", so you can't set them equal to each other.