Prior to 1916 there was no official design, says the flag website nava.org:

Q: I have two different 48 state flags. One has 48 staggered stars and the other has eight equal rows across with six rows down. Were there two different 48 state flags ever made? - Marla

A: Prior to 1916, there were no official specifications for US flags. Indeed these two variants show up right from the beginning of the 48 star flag in 1912. After 1916, pretty much everyone switched to the even rows with the stars one over the other like the second one you mention. 48 star flags with staggered rows are somewhat rare and are early (1912-1916 or thereabouts).

…the staggered rows, not just the stars in the frame of the photo. I guess a 48-star flag was still new enough that perhaps the maker of it hadn't seen the official non-staggered 6-rows-of-8 version.

[Prior to 1916 there was no official version. See comment above. - Dave]

There is a man, just right of the very center of this photo who appears to be aiming a machine gun at a hole in the ground. It looks like a group of people have gathered around to watch. His white pants would indicate that he is a custodian. Hmmm.

[Diabolical. And ingenious. His machine gun is cunningly disguised as a board. - Dave]

If this picture was taken in 1913, I'll eat my hat. My bowler hat. Or my wife's Merry Widow hat. Nobody dressed like this in 1913. Not a hobble skirt in sight. This photo is from 1908-1910 and no later. Trust me, I know.

[The date is written right on the glass. Plus the flags have 48 stars. Bon appetit. - Dave]

...and it still looks like that to this day.

…or is that upper left flag a 45-star flag? Unless this was some sort of old flag display (which it doesn't really appear to be), they should have been flying a 48-star flag. It was adopted on July 4, 1912 after Arizona and New Mexico were admitted to the union.

[It shows 38, not 45. But it has 48. Six staggered rows of eight stars each. One corner is out of the frame, so we can't see them all. If you look at the flag on the right, you will see it also has 48 stars. Eight stars in the top row, eight stars in the second row, etc. - Dave]