You need to read the passage in the following way:
If you know your space animals, then you're probably already familiar with Ham, the first chimp America ever launched into space. You also probably know Laika, the late, great Russian space dog and the first animal to orbit Earth.
This is because late, great is a fixed expression that means "remarkable but no longer with us." You need to take the expression as a whole and not try to plug in individual synonyms so that you come up with something like
You also probably know Laika, the deceased, remarkable Russian space dog...
In this case, yes, the two adjectives are used in a way that each one individually applies. The two adjectives are also not a set phrase or set expression.
With late, great each adjective does apply, but as a fixed pair as part of a fixed expression. The meaning is taken as a whole, which is
You probably also know Laika, the remarkable but no longer with us space dog...
That is the meaning taken as a whole.
And since the phrase is applied to deceased persons (persons here including dogs and mammals), you can actually think of the phrase as simply meaning famous.
And to call Laika famous does not imply anything about Ham. The passage is about two famous space mammals. The passage presumes that "know your space animals" and that "you're probably already familiar with Ham" and that "You also probably know Laika."
But saying "late, great" with regard to Laika is not emphasizing that Laika is dead. Poor dog! No! It is calling her "remarkable but no longer with us." Ie, "famous." And the writer presupposes that we already know this. (If we do not already know this, then the set phrase, taken as a whole, indirectly tells us that Laika is deceased.)
By saying "late, great" ("remarkable but no longer with us"), it is not implying anything about Ham. Readers are meant to know that Ham, as well as Laika is dead--and remarkable.
If the passage said that Laika was "brown" or "partially brown," it would not be emphasizing that Laika was partially brown and that the monkey was not brown. The passage presupposes that the reader knows that monkeys are also partially brown. Just like the reader already knows that Ham, as well as Laika, are deceased. It is simply calling Laika famous.
Moreover, a person almost always uses the phrase "late, great" in reference to a person that 'everybody' (as in 'people in general') knows is deceased. In this way, it is used as a reminder or as an homage to the deceased person.
If I say, "the late, great Elvis" I expect you to already know a) who Elvis is, b) that he is dead, c) that he is famous. I am not informing you of any of these things. I am simply calling Elvis "remarkable but no longer with us." In this way, we call to my mind famous or well-known people who have passed on.
But a person can refer to someone that is not universally famous. In this case, the person is probably going to tell you more about him.
If I say, "the late, great Lefty Grove", I expect that you probably do not know a) who Lefty Grove is, but I am immediately informing you b) that he is dead and c) that he is remarkable. So you should prick up your ears, like Laika would, because I am probably next going to tell you who he was and why he was remarkable.
Furthermore, because the words rhyme, the words late and great are often used together in another expression: the latest and greatest. This is used to describe a most recent version of a product or idea that is supposed to be the best, or better than previous versions.
Although the free dictionary does not give a definition on this page, it lists a lot of usages of late and great together at the bottom. You can also find plenty of examples if you just Google "late, great".