Is there any monitor that can send the DPI info to the computer using Display Data Channel or similar?
You are right, monitors do not send DPI information.
But EDID (Extended Display identification Data), which is transmitted via DDC (Display Data Channel), contains both the timing information (the resolution) as well as the physical screen size, so software can calculate the monitor's DPI (or better: its pixel density, normally measured in PPI) using these values.
The display's rough physical size information is represented in EDID as Horizontal Size (in centimeters, at offset 21) and Vertical Size (in centimeters, at offset 22) and, more precise (in millimeters), in the Detailed Timing Descriptor Blocks at offsets 12 to 14 (Horizontal Display Size and Vertical Display Size).
The display's resolution is stored in the Detailed Timing Descriptor Blocks, beginning at offset 54 (Horizontal Active Pixels and Vertical Active Lines).
One can expect that the resolution information is correctly programmed in all monitors because this value is used by the operating system to correctly set the screen resolution. The size information however can be wrong or even completely missing because it is not a hard requirement (or not even needed) for setting the screen resolution correctly.