I would actually encourage you to see the poem as presenting the opposite idea: that the brook in this poem maintains its identity, asserting itself again and again as something permanent: "For men may come and men may go, / But I go on for ever."
While it's true that the brook shows itself as often changing, sometimes chattering and other times murmuring, sometimes winding and sometimes leaping forward in "a sudden sally," the often-repeated refrain of the poem seems like a very clear insistence of the notion of the brook as a constant, steady entity with a clear and lasting identity.
If you wanted to argue that the brook does lose its identity, you might focus on the fact that it goes to join the river over and over, and yet it stays a brook. You might interpret this repeated confluence with the larger body of water as multiple endings of the brook's particular journeys. However, this argument is rather untenable. You'll find yourself trying to answer the question of whether the brook is an abstract notion of a brook or whether it is actually the individual set of water molecules that flow from a starting point ("haunts of coot and hern") to an end point ("the brimming river"). That question is a bit tangential to the poem's purpose, which is to explore how the permanent identity and character of the brook provides a humbling juxtaposition to the brevity of human life.