My apologies - for some reason I saw that as one big URL...
Since you have the two components jointed, you can in fact cut out the servomotor from the bracket with a Combine operation. However, that probably won't give you the kind of clean hole you really want since it will include extra details such as the motor's fillets. Also, it yields only an exact fit with no wiggle room.
I would suggest taking the time to learn more about sketching, projections, and extruding in Fusion 360 since these will let you design exactly the kind of hole that you want.
Below is an example that uses the Intersect operation in a sketch to capture the rectangular profile of the motor and remove it from the bracket through extrusion. The projected profile is offset by 0.1mm to allow for some fitting slop, and the offset profile is then used to size a rectangle used for the cutting operation. (If you're OK with preserving the fillets on the actual motor profile, you don't need to do that last part. Just cut with the offset profile.)
If you want a more exact fit where the rectangular cutout goes only halfway through the bracket, and you add a separate through-hold for the spindle, you can elaborate on this same basic methodology.
I suspect that the problems you referred to with your earlier attempt at projection stemmed from trying to project the silhouette of the spindle. For some reason, the flat sides double-project into the sketch and don't create a well-formed profile. (It might be that the spindle isn't properly aligned to the model coordinate system.) To get around that issue, just project the top outer edge of the spindle and alignment bump instead of using the flat sides.
