The speaker compares herself to a mountain silhouette instead of the more alluring image of a beautiful woman in a dim room. Mountains are impressive at sunrise, but people soon stop noticing them. That is how she feels in someone's life: present, dependable, and gradually ignored.
In the second stanza she imagines being the other kind of silhouette, one that could hold attention longer than "two minutes" and win the closeness and affection she wants. The tone is wistful and a little self-doubting, since she wonders whether she has to prove she is worthy. The puzzle piece image deepens the loneliness: she feels she belongs to a different box, and the other pieces have been scattered so no one can find the way back. The last line, "Mountains never move," is ambiguous. It can sound like quiet resignation, but it can also read as stubborn strength. Overall the mood is sad and tender, with a thread of endurance.