At some point, questions like "what makes this word mean what it does" are not very intelligible; you have to accept that eventually, except in cases where a word can be linked to onomatapoeia or something, that you're going to wind up at a moment of aporia.
In this case however, we can possibly say something intelligible about why "all others" means "everyone who is not in [defined group A]", and why "every other" means "one out of two, repeated serially". It has to do with the difference between "all" and "every" as expressions of quantity, and what they mean literally in terms of collectivity.
"All" and "every" both refer to a collective whole. "All" refers to the group as a plural entity - that is, an entity in which the constituent members lose their individuality.
In this sample sentence, note that the noun "dogs" is plural, and conjugates with the plural verb "have". This is because the dogs have lost their individuality in this case. The speaker is not considering the plight of the individual dog, but is rather making a statement about dogs as a whole.
Let's look at "every" now. This word also refers to a plurality, but it is one in which the constituent members retain an individuality.
The noun is now "dog", and the verb is the singular "has". This is important, as we are now thinking of the dogs in this group on a more individual level.
Back to your original question. "All others" means "a non-individuated plurality of others" - which, to me, sounds like "all [people, etc.] who are not in [group A]". If we look at "every other", well, now we have to do some mental gymnastics. Imagine that you have 10 people in a line. Starting with person number one, you arbitrarily assign them an "other" - that is, a counterpart who is somehow different. To do so, you'd take the next person in line - number two. That gives us two people who have an assigned identity - number one being a "normal" and number two being an "other". We move onto the next person in line; number three is then assigned an "other" - who, for convenience's sake, is number four - now we're up to four assigned identities - one and three are "normals", while two and four are "others".
Eventually, once you have assigned "every other", you are left with a series in which the second person out of two in the series is marked as "other". Then you can, oh, I dunno, give them swords and make them fight gladiatorial battles for you. It's your thought experiment.
