...but if there's a newline, then it isn't the last line, is it?

Newline is traditionally a line terminator not a line separator. But even if you consider newline a line separator, then it is still a bit of a stretch to call the 0 characters after the final newline "a line". I think you have been mislead by quirks of notepad (or its kin) which shows space for a line below the last line since that space is where the cursor goes when you press the final ENTER.

Certainly, <> won't read "a line" after a final newline in a file (it won't read anything, other than "end of file"). Now, if the last newline in a file is followed by more characters, then the historical view would be to consider that "a partial line" or "an unterminated line" or similar terminology. Traditionally, vi would refuse to create such files and if you used vi to edit such a file and saved any changes, vi would add a final newline silently (this has changed with newer vi replacements like, for example, vim).

It would be better expressed as "the terminating identifier isn't allowed to be the last line".

Better to change the verbage to not rely on either view of what "last line" means. Perhaps:

If the terminating identifier is at the very end of the program without even a newline after it, then Perl will give the warning Can't find string terminator "END" anywhere before EOF. So be sure to include a newline immediately after the terminating identifier.