System Information
Operating system: macOS-10.15.7-x86_64-i386-64bit 64 Bits

Blender Version
Broken: version: 3.5.0 Alpha, branch: master, commit date: 2022-12-02 21:53, hash: rB9719fd696488

Short description of error
I followed the procedure using this video:
https://www.youtube.com/watch?v=gz4qV5_zuMo

And when painting the texture from the image this is what I've got:

Went to add some edge loops to increase subdivision for accuracy:

Then return to painting and this is what I got:

This is the image I have downloaded and use for example:
https://media.istockphoto.com/id/173803871/photo/any-time-road-sign.jpg?s=612x612&w=is&k=20&c=9-K5mZ7tNOiMklmRNkBNIvy-9AZfk4sSCiTdDbD3aZY=

This is the file I am using:

Video capturing creation of error, sorry I have to speed up to reduce the file size:

Reproducing steps

• Start Blender, remove default cube
• Press numpad 0 to camera view
• Select Camera tab from Properties
• Select Camera's Background Images
• Add Image, select the no_parking_sign.jpg

• Load the no_parking_sign.jpg into Gimp, and find the dimension in Image Properties (612 x 408px)
• In Scene tab, Format panel, enter Resolution X: 6120, y:4080
• Shift-A to add a Plane, turn on the Axis so you can follow and understand the reason for UV map going wrong later.

• Tab into Edit Mode and then click on Viewport Overlay, turn on the Hidden Wire to remove the solid face while you're performing next operation.

• Notice the +Y and -X direction of the Axis of the plane as you have added the plane in 'World' Align, not 'View' Align. (This is where things going wrong, UV Map should have '+Y' pointing up)
• Grab two points off -X axis to match up the two top corners of the 'No Parking' sign on the camera background image
• Grab two points off +X axis to match up the bottom two corners of the 'No Parking' sign on the camera background image
• Switch to UV Editing workspace.
• Select the default UV Map in the UV Maps tab in the properties and in the UV Editor press 'S, X' to scale the default UV in X axis, to an equivalent proportion of the parking sign appear on the background image.
• Edit UV Maps tab in the Properties panel, rename the default 'UV Map' as 'original'.
• Click + to add a new UV Map and rename it as 'projected'.

• While 'projected' UV map is selected, press U on the plane in Edit Mode, choose to unwrap as 'Project From View'

• Switch to Texture Painting workspace.
• Select Texture Slots in Active Tool and Work Space settings, select Mode as Single Image, click 'New' to add a new image, set Name as 'extracted_texture', leave default width and height as 1024, and background color as Black, click OK.

• Select 'original' in the Active UV Map index (0), and 'Linear' interpolation
• Select the Clone Brush on the tool panel on right
• In the Brush Settings tab, on Properties panel, check the 'Clone from Paint Slot'
• Set 'Source Clone Image' as the 'no_parking_sign.jpg'
• Set 'Souce Clone UV Map' as 'projected'

• Now using the clone brush and draw on the black plane on background image of the camera to clone it to the 'extracted texture' on the UV Editor.
• You will notice that the result UV map projected is rotated.

• The file I'm using:

• If you redo the same set of steps, but after added the plane in, at the 'Add Plane' panel, Align field, choose 'View' instead of 'World' default, and the final result would be as expected.

As found out, there should be somewhere in the UV editing warning of this fact, that +Y of the plane is expected?