Image's blend mode is set as multiply and it is showing the effect correctly in Adobe Fireworks. However, when overlapping the same image with another, separate image in the browser, the effect looks different.
The standard JPG/GIF/PNG files Fireworks exports do not support blending modes.
If you blend two images in Fireworks together, it doesn't export two images with blending modes, but rather than one blended image. If you have only one image with a blending mode set, Fireworks exports the image blended with the background (or any other layer(s) behind it). The result is not dynamic and that's why it works as you described.
There is some related discussion on Stack Overflow too: Are photoshop-like blend modes possible in HTML5?