In Echad Mi Yodea song both #11 and #12 refer to the tribes - without Yosef and with him. This is especially pronounced in the families that go through the full list on every line, like "the 3 forefathers - Avrohom, Itzchok and Yaakov".
Why they both refer to [almost] the same thing?
(This answer tries to explain eleven, but my question is why those are the same thing)
The eleven refer to the stars of Yosef's dream rather than all of the tribes.
אַחַד עָשָׂר מִי יוֹדֵעַ? אַחַד עָשָׂר אֲנִי יוֹדֵעַ: אַחַד עַָשָׂר כּוֹכְבַיָּא
Twelve on the other hand refers to the actual tribes.
שְׁנֵים עָשָׂר מִי יודע? שנים עָשָׂר אֲנִי יוֹדֵעַ: שְׁנֵים עֶָשָׂר שִׁבְטַיָא,
While the interpretation of the dream says that the eleven stars referred to the brothers, the actual reference is to the stars that were visible in the dream. Thus, it is not the same as the twelve.
As per the Oz V’hadar extended Mesivta Haggadah, page 826:
The Sefer Leil Shimmurim (by Rabbi Yechiel Epstein, author of the Aruch HaShulchan) writes that the eleven stars correspond to the seven “planets” (sun, moon, Mercury, Mars, Jupiter, Venus, and Saturn) plus the four elements (earth, fire, water, and air). Similarly the Bnei Yissaschar writes that they correspond to the eleven “planets” (it’s unclear to what he refers, though this is discussed by Ibn Ezra to Tehillim 8:5 as well).
I have not tried to research this excellent question, but my initial thought is that it follows the dual ritual of the Seder- remembering the slavery AND remembering the freedom. For example: Matzoh is BOTH poor man's bread (representing the slavery) AND freedom bread (not enough time to bake). Many of the other rituals refer to one or the other (or both).
In essence the Seder is recalling both slavery and freedom. 11 stars represents slavery by alluding to the 11 brothers coming down to Egypt (Joseph was already there). 12 represents the freedom, were we became a nation of 12 tribes after we were redeemed.
