What is the best way to mount a flag on shake (shingle) siding?
You need to find framing members to screw the bracket to (studs) most homes built in this era have a sub siding and then the exterior siding.
For the best strength close to a corner or window frame may provide multiple studs the width of your bracket, if not a single stud on one side with the other side screwed into the siding. The side that you connect to a stud we would normally use lag bolts 1/4” is a common size but the length varies depending on the load being applied , for a 6’ pole and flag like we had on my upper deck I did use 1/4” x3” lag screws I only had a stud on 1 side of the bracket so the other side I used 1-1/2 ‘ screws, this area gets strong wind so I put the bracket over the stud on the leading side.

To explain looking at the house the wind almost always blows from the left to right. 
I put the bracket as far to the right to allow the left holes to line up with the stud. 
This way most of the force from the wind is pulling on the strongest point of attachment.

If you have strong winds in both directions closer to a corner or window where there are normally multiple studs closer together may provide the support.