I see the poem "On the Beach at Night Alone" speaking of patterns and cycles. Those, in turn, are symbolic of the sameness that unites everything through that repeating motion that is life.
Whitman counts down "everything" in his poem. From smaller objects to the bigger, from concepts to specifics to parallels. They are all, he says, connected by "a vast similitude." You could say it means we are all the same — being made of star stuff, like Carl Sagan said. But it's also possible to interpret it meaning that we share the same fate, or travel together on the same eternal path as every single thing with a beginning and an end. A person is born and dies, just the same as a language would. Or fish, or nations, or even suns. Between those moments, however, we have our own personal quirks and individuality; but there is also a "similitude" that links us to other beings who are the same.
There is a line in the poem that illustrates this well, I believe:
As the old mother sways her to and fro singing her husky song
There is something very ancient and eternal about this. A mother singing to her child is far from being a lost custom — in fact, it could be the opposite. It's a happening that we can imagine taking place as far as history reaches — in both directions. You could consider the song itself, too. We think of lullaby as something with a history: passed from mother to daughter, forming a link through generations. Thus it could be symbolic of everything else, repeating throughout time. All different, but all the same.