There is a bug in the Fusion 360 .obj texture handling I presume?
When opening .obj file the .obj file is linked to equally named .mtl file in the same folder. The .mtl file is a text file that includes the information about the texture .png file that is supposed to be used over the .obj file. The .jpg file is located in the assets folder.
If the obj. jpg file has same name for example color.jpg and then you open new .obj it can happen that Fusion 360 still opens the old texture jpg file over the new object because it has same name.
Workaround: go to .mtl file (open with text editor) it can look something like this:
newmtl Material
map_Kd assets/color.jpg
Pr 1.00
Now change the name of the color.jpg to color2.jpg so that it looks like this:
newmtl Material
map_Kd assets/color2.jpg
Pr 1.0
Then also change the color.jpg name in the assets folder to color2.jpg
Now the texture opens normally again.
This is a probably a memory BUG that Fusion 360 has. Autodesk should fix it immediately if this is the case.