To select the fragment we should know start and end drag points and get text positions for all the y coordinates in the rectangle. After that we convert them into model offsets via viewToModel method and add custom highlights. See the example app class below.

import java.awt.*;
import java.awt.event.*;
import java.awt.datatransfer.*;
 
import javax.swing.*;
import javax.swing.text.*;

public class VerticalHighlight {
    JFrame frame;
    JTextArea ta;
 
    public static void main(String args[]) {
        new VerticalHighlight();
    }
 
    public VerticalHighlight() {
        frame = new JFrame();
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        ta=new JTextArea("test11111\ntest22222\ntest33333\ntest44444\ntest55555\n");
        ta.setCaret(new MyCaret());
        JScrollPane scroll=new JScrollPane(ta);
        frame.getContentPane().setLayout(new BorderLayout());
        frame.getContentPane().add(scroll,BorderLayout.CENTER);

        JButton copy=new JButton("Copy");
        ActionListener lst=new ActionListener() {
            public void actionPerformed(ActionEvent e) {
                try {
                    Highlighter.Highlight[] selections= ta.getHighlighter().getHighlights();
                    String text="";
                    int cnt=selections.length;
                    for (int i=0; i<cnt; i++) {
                        int start=selections[i].getStartOffset();
                        int end=selections[i].getEndOffset();
                        String selectedText=ta.getDocument().getText(start,end-start);
                        text+=selectedText+'\n';
                        System.err.println(selectedText);
                    }
                    StringSelection ss=new StringSelection(text);
                    Toolkit.getDefaultToolkit().getSystemClipboard().setContents(ss,ss);
                }

                catch (Exception ex) {
                    ex.printStackTrace();
                }
            }
        };
        copy.addActionListener(lst);
        frame.getContentPane().add(copy,BorderLayout.SOUTH);
        frame.setSize(300,200);
        frame.setLocationRelativeTo(null);
        frame.setVisible(true);
    }
}
 
class MyCaret extends DefaultCaret {
    Point lastPoint=new Point(0,0);
    public void mouseMoved(MouseEvent e) {
        super.mouseMoved(e);
        lastPoint=new Point(e.getX(),e.getY());
    }
    public void mouseClicked(MouseEvent e) {
        super.mouseClicked(e);
        getComponent().getHighlighter().removeAllHighlights();
    }

    protected void moveCaret(MouseEvent e) {
        Point pt = new Point(e.getX(), e.getY());
        Position.Bias[] biasRet = new Position.Bias[1];
        int pos = getComponent().getUI().viewToModel(getComponent(), pt, biasRet);
        if(biasRet[0] == null)
            biasRet[0] = Position.Bias.Forward;
        if (pos >= 0) {
            setDot(pos);
            Point start=new Point(Math.min(lastPoint.x,pt.x),Math.min(lastPoint.y,pt.y));
            Point end=new Point(Math.max(lastPoint.x,pt.x),Math.max(lastPoint.y,pt.y));
            customHighlight(start,end);
        }
    }
 
    protected void customHighlight(Point start, Point end) {
        getComponent().getHighlighter().removeAllHighlights();
        int y=start.y;
        int firstX=start.x;
        int lastX=end.x;
 
        int pos1 = getComponent().getUI().viewToModel(getComponent(), new Point(firstX,y));
        int pos2 = getComponent().getUI().viewToModel(getComponent(), new Point(lastX,y));
        try {
            getComponent().getHighlighter().addHighlight(pos1,pos2,
                     ((DefaultHighlighter)getComponent().getHighlighter()).DefaultPainter);
        }
        catch (Exception ex) {
            ex.printStackTrace();
        }
        y++;
        while (y<end.y) {
            int pos1new = getComponent().getUI().viewToModel(getComponent(), new Point(firstX,y));
            int pos2new = getComponent().getUI().viewToModel(getComponent(), new Point(lastX,y));
            if (pos1!=pos1new)  {
                pos1=pos1new;
                pos2=pos2new;
                try {
                    getComponent().getHighlighter().addHighlight(pos1,pos2,
                             ((DefaultHighlighter)getComponent().getHighlighter()).DefaultPainter);
                }
                catch (Exception ex) {
                    ex.printStackTrace();
                }
            }
            y++;
        }
    }
}