One possible workaround (for the pdf backend) is to explicitly set the dpi of the figure, i.e., put this line of code directly after the figure creation with the plt.subplots() call:

fig.dpi = 72

So, this workaround confirms that the fig.dpi setting is not handled properly in the transformation of the BoundingBox.

Note that this has nothing to do with the keyword argument dpi in the call to text.get_window_extent(). Specifying this with different values changes the anchor point of the BoundingBox, thus shifting it alltogether. But we need to only change width and height.

Note further that changing the fig.dpi changes width and height of the BoundingBox for both, the PS and the PDF backend. But it does not do so for the AGG backend and PNG output. I think, I'm still missing something ...

Thanks for the clear and detailed report.

I think the real problem here is that you cannot get the extention of a text without rendering it first; but rendering requires a DPI, even for backends that are pixel-independent in their output.
As you have seen you need to actually save the image first, in order to be able to use .get_window_extent(), because that requires a renderer. So with that renderer you get the bounding box of the text in pixel space. And pixel space for the pdf backend is always defined via a dpi of 72.

A workaround which avoids the dpi coming out of sync would be to use the agg backend throughout and only saving the figure via the respective vector backend:

import matplotlib as mpl
mpl.use("agg")
import matplotlib.pyplot as plt

fig, ax = plt.subplots()
ax.plot([0, 1], [1, 0], "bo", ms=20)

text = ax.text(x=0.3, y=0.1, s="sample text", fontsize=20, family="Serif")

fig.canvas.draw()
bbox_data = text.get_window_extent().inverse_transformed(ax.transData)

rect_data = mpl.patches.Rectangle(
    (bbox_data.x0, bbox_data.y0), bbox_data.width, bbox_data.height,
    fill=False, lw=1, color="r")
ax.add_patch(rect_data)

fig.savefig("bbox.png")
fig.savefig("bbox.pdf")
fig.savefig("bbox.ps")

I'm not sure what you're trying to do here, but if this is strictly what you're looking for (and not just some simplified case), a simpler workaround is to use the bbox parameter. That is:

import matplotlib as mpl
mpl.use("PDF")  # noqa

import matplotlib.pyplot as plt
import numpy as np

fig, ax = plt.subplots()
ax.plot([0, 1], [1, 0], "bo", ms=20)

text = ax.text(x=0.3, y=0.1, s="sample text", fontsize=20, family="Serif",
               bbox={"fill": False, "lw": 0.2, "color": "r"})

fig.savefig("bbox2.pdf")

The default boxstyle is square,pad=0.3 so if you want it flush, then pass a style with an explicit padding:

text = ax.text(x=0.3, y=0.1, s="sample text", fontsize=20, family="Serif",
               bbox={"fill": False, "lw": 0.2, "color": "r",
                     "boxstyle": "square,pad=0"})

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