How to remove day grouping header for today's date in Agenda View?

You can define org-agenda-format-date either as a string or as a function to format the date in the agenda. C-h v org-agenda-format-date RET says:

Format string for displaying dates in the agenda.

Used by the daily/weekly agenda. This should be a format string understood by ‘format-time-string’, or a function returning the formatted date as a string. The function must take a single argument, a calendar-style date list like (month day year).

So all you need to do is add a setting for it in your org-agenda-custom-commands:

(setq org-agenda-custom-commands
        '(("o" "Custom Agenda"
           ((agenda "" ((org-agenda-overriding-header "n  ⚡ Do Today ⚡n⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺")
                    (org-agenda-remove-tags t)
                    (tags-todo "today")
                    (category-todo "today")
                    (org-agenda-highlight-todo)
                    (org-agenda-span 'day)
                    (org-agenda-skip-deadline-if-done)
                    (org-agenda-time-grid nil)
                    (org-agenda-format-date "")  ; <---- HERE
                    (org-agenda-prefix-format "  %-3i  %10b %15t%10s")))))))