Thanks for the detailed steps, Coises — that Shift+Right Arrow trick to turn the column selection into a multi-editing selection is the part I had been missing.
One alternative that worked well for me in a similar case, when the values are already on separate lines, is to skip the cut step altogether: place a caret at the end of each target line with Ctrl+click (multi-editing), then paste the multi-line text. Notepad++ puts one pasted line at each caret, so the 1 goes after the first ID = and the 2 after the second, with no blank lines left behind. It is a bit fiddly if there are many lines, but for two or three lines it is quicker than converting the selection back and forth.
Either way, checking that both multi-editing options are enabled in Settings > Preferences > Editing 2 first, as you said, seems to be the key.