There has been some discussion of who the lady found here is, but the general concensus today is with Magdalena. It is also highly unlikely that any further light will be cast on this painting, with so many centuries having past since the artist passed away.
Whilst the style is typical of Rembrandt's late work, there are some elements to this painting which slightly differ from the norm. For example, animals were very rarely seen within his career, perhaps meaning that in this case the model had a strong bond with her pet dog that she wished to be communicated in the painting.
The amount of light used is also significantly more in this artwork, when normally the artist may just bring out the facial features and leave the torso still relatively dark. This may have been the request of whoever commissioned the piece, or perhaps Rembrandt himself wished to draw particular attention to the attractive figure of his model.
The background to this scene remains neutral and bland, with the artist ensuring that no element of it took one's attention. The young lady herself looks pensively across, perhaps suggesting that she had not stood for the artist before. Perhaps she was not comfortable in such grand clothing, or was intimidated by the master's artistic reputation.
This painting from 1665 can be found at the Art Gallery of Ontario in Canada.