You haven't a command in Illustrator "recolor the background of this image to..." because Illustrator hasn't any idea which white parts of the photo belong to the background.
Recolor the background in a photo editor
Remove the background of the photo (=make transparent) in a photo editor
Insert in Illustrator an opacity or clipping mask or a shape which covers the unwanted white
Option 2 is the standard way because it lefts most freedom. You must save the photo in a format which supports transparency, for ex. PNG or PSD, but not JPG.
There exists no general automatic procedure for any of the options, they need manual work which is often a detailed path drawing along an edge. Only in some lucky cases you can make succesful selections with magic wand or by color in Photoshop or trace the photo in Illustrator and separate the shape for the needed mask.
Of course there's blending modes. You can place your photo with blending mode multiply. Set the mode in the transparency or appearance panel. If your photo happens to be pure greyscale image, that will work perfectly. Generally the colors get distorted with blending modes.