Poems are usually written as a way to express the feelings of the author. The person or place or event that has created those feelings is the subject or object of the poem. There isn't one uniform subject or object (different names for the same thing) in all poetry - it will be different for each poem.
Lyric poetry is "a form of poetry with rhyming schemes that express personal and emotional feelings." Words don't have to rhyme for a piece to be considered a lyric poem, although they can. The more important issue is the expression of feelings. Consider:I THINK that I shall never see A poem lovely as a tree. A tree whose hungry mouth is prest Against the sweet earth's flowing breast; A tree that looks at God all day, 5 And lifts her leafy arms to pray; A tree that may in summer wear A nest of robins in her hair; Upon whose bosom snow has lain; Who intimately lives with rain. 10 Poems are made by fools like me, But only God can make a tree.
Joyce Kilmer is observing the beauty she sees in trees, admiring the strength of the trunk growing from the ground, appreciating the sight and sound of the robins nesting amidst the leaves, recognizing the way the tree changes with the changing seasons. The lyric subject or object of this particular poem is trees.