Is this not a very self-serving assertion, though? Pointing at things that we humans have accomplished and are proud of, and saying that sets us apart, or are in a sense categorically higher than other animals?

Other animals may well look at this and look down at us for having to do all these things in order to feel accomplished. Octopuses may draw a qualitative distinction between them and us because we lack the means to alter our appearance--surely they're more physically manifest than other animals due to their color shifting abilities! Plants must belittle us for our inability to passively absorb energy from the sun.

We could definitely say that our ability to get to the moon sets us apart. But in doing so, we also have to acknowledge each and every other trait that would set an animal species apart from the rest, and there's no shortage of those unique elements.