The main problem with your code is that you create a new PhotoImage for each pixel! Instead, create the PhotoImage once and just add the pixels in the double-for-loop.

def box(event):
    yaxis(event)
    canvas.create_rectangle(x1, y1, x2, y2)

    picture = PhotoImage(width=(x2-x1), height=(y2-y1))
    for x in range(x1, x2):
        for y in range(y1, y2):
            r, g, b = photo.get(x, y)
            picture.put("#%02x%02x%02x" % (r, g, b), (x-x1, y-y1))
    picture.write('new_image.gif', format='gif')

Also, the line tuple(map(int, value.split(" "))) in your getRGB function is wrong, as value is already the tuple you want to create, not a string.¹⁾ As you can see, I just 'inlined' that part directly into the box function. Another problem was that you wrote the copied pixels to x and y, but you have to write them to x-x1 and y-y1 instead.

Update 1: ¹⁾ It seems like the return value of PhotoImage.get depends on the version of Python/Tkinter you are using. In some versions, it returns a tuple, like (41, 68, 151), and in others, a string, like u'41 68 151'.

Update 2: As pointed out by @Oblivion, you can in fact just use the from_coords parameter of PhotoImage.write to specify the region of the picture to be saved to file. With this, the box function can be simplified as

def box(event):
    yaxis(event)
    canvas.create_rectangle(x1, y1, x2, y2)
    photo.write('new_image.gif', format='gif', from_coords=[x1, y1, x2, y2])

I know that it might sound totally unrelated but I've recently had a similar issue. Thing is, I'm on Linux and can't check on Mac. Bash defaults changed in 5.1, that was the root cause. In my case the text was also pasted 'highlighted' when I pasted it in regular shell, and it turned out that setting enable-bracketed-paste off helped - not only with the highlited thing but also with the python3 shell paste, which was broken for some time as well.

So, I'd recommend having a look into some input defaults / settings, on Linux distros that can be configured in the .inputrc file.