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The problem was that the Sage uses additional scaling coefficient to font size of labels called axes_labels_size, setting it to 1 and using fontsize directive resolves the problem:

p = plot(sin, 0, 2 * pi,  axes_labels=('$x$', '$y$'))
p.save(file_name)

becomes

p = plot(sin, 0, 2 * pi,  axes_labels=('$x$', '$y$'), axes_labels_size=1, fontsize=12)
p.save(file_name)