There is no known מקור for this הלכה
It is very common on Friday night for someone who didn't finish
his שמונה עשרה with the ציבור to seek a partner to say with
This is based on the משנה ברורה רס"ח ס"ק י"ט who writes
"If ויכולו wasn't said with the ציבור then at least it should be
said with a second person".
The Klauzenburger Rebbe Zt"l said that this Minhag was
unheard of in the old country.
The Chazon Ish argues on the משנ"ב and claims no need to do so
since we only find a הידור to do so with ten and not with two.
This Minhag of the משנ"ב has no Mokor in any of the Poskim
Rishonim or Achronim. It is the משנ"ב who was מחדש this
Halacha and yet today it is accepted in all of כלל ישראל Sefardim,
Ashkenazim, Chasidim and Misnagdim.
According to the אריז"ל, you are
supposed to say on Friday
night ויכולו only three times, not more and not less.
(when partnering you will be saying it 4 times unless you
don't make Kiddush)