Within the context of the poem
As she had changed her kind
has the meaning that she has changed her state of being. She has become a mother as stated in the previous stanza
Four ivory eggs soon pave its [nest] floor
The scene is on a ship and the two birds have used the ship's mast as their tree to nest in.
The mother-bird is gone to sea
The mother bird, while incubating her eggs must go wherever the nest goes, which is tied to the mast, which is attached to the boat, which has set sail. She goes because she is a mother now with eggs (so to speak), and her existence becomes more of a sea bird than the land bird that she actually is, which is also a change in kind for the mother-bird
This is contrasted with the father-bird
But goes the male? Far wiser he
Is doubtless left behind.
The expectation is he will stay behind because chaffinches are land birds, not sea birds, therein lies his wisdom to stay behind on land. However
He flew to reach it [the nest], by a law
Of never-failing love.
because of his love for his mate he joins her, and a reference to the danger involved is made later
The billows and the blasts defied
Were not afraid to plough the brine
In company with man;
They would normally be fearful to be so far off-shore since they are land birds.
It is about a mother's instinctive duty to her young, and a husband's love for his mate, and staying together despite the odds. Seeing this, the sailor's view it as a favourable sign.