It is stated that a convert is in the same regard considered a Jew as a native born. Rashi on Dvarim 26:11 states that a convert cannot recite the declaration כי באתי אל הארץ נשמע ה׳ לאבתינו לתת לנו. Why is that? At conversion a convert receives a Jewish neshama, thus making him part of the Jewish nation, a Jew in every respect. Rabbi Manis Friedman said in a shiur that the reason a converts needs to convert is exactly for the reason of becoming a Jew in every regard. When he receives a Jewish neshama now he is 100% a Jew, and he can say "our fathers" etc. Here is part of the shiur: https://www.youtube.com/watch?v=UZMX7Y7ava0
You're absolutely right to compare this statement to saying "our fathers" in tefillah. The mishnah in Bikkurim 1:4 makes the same comparison and rules that a convert can't say either one.
We don't pasken like that mishnah. Instead, we follow Rabbi Yehuda, quoted in a braisa in the Yerushalmi, who rules that a convert does say "our fathers" in Shemoneh Esrei and will be able to say mikra bikkurim as soon as we have the Beis Hamikdash operating properly.
I also disagree with the premise of your question. Everyone agrees that converts are fully Jewish, but there are certain statements that converts can't make. For instance, a convert can't say "I am not a convert" or "I am a biological descendant of Avraham". The question is how closely the word "our fathers" is tied to biological descent.
The Rambam allowed a convert to say “God of our fathers” in prayers, which a convert was, up to that point, not allowed to say, because his “fathers” were not Jewish. Indeed, the Mishna in Bikkurim does not allow it. [The Babylonian Talmud (the Bavli) has no Gemara in Bikkurim]:
The convert brings [first fruits] but does not recite [the relevant declaration in Deuteronomy] because he cannot say “the land which God has sworn to our fathers to give us.” [Deuteronomy 26:3]. But if his mother was of Israel, he may [both] bring [first fruits] and recite [the declaration]. And when he prays by himself, he says, “God of the fathers of Israel”. And when he is in synagogue [with the community], he says, “God of YOUR fathers.” But if his mother was of Israel, he says, “God of our fathers.” [B. Bikkurim 1:4]
But the Jerusalem Talmud (the Yerushalmi) allows it:
A… convert brings [first fruits] AND recites [the declaration in Deuteronomy]. Why? [Because God said to Abraham]: “For I have made you the father of a multitude of nations” [Genesis 17:5]. In the past you were a father to Aram [only], but now, henceforth, you are a father to all the nations. [J. Bikkurim 1:4, 64a]
Maimonides ruled in favor of the Yerushalmi. In his letter to a convert named Obadiah, he says:
You ask me if you, too, are allowed to say… “Our God and God of our fathers”, “who has sanctified us through His commandments”, “who has separated us”, “who has chosen us”, “who has given to our fathers to inherit [a pleasant, good and spacious land]”, “who has brought us out of the Land of Egypt”, “who has worked miracles to our fathers”, and more of this kind. Yes, you may say all this in the prescribed order and not change it in the least…
And he concludes:
Toward father and mother we are commanded to honor and revere them, toward the prophets to obey them, but toward converts we are commanded to have great love in our inmost hearts.... God, in His glory, loves a convert --... [because he is] a man who left his father and birthplace and the realm of his people at a time when they are powerful, who understood with his insight, and who attached himself to this nation which today is a despised people, the slave of rulers, and recognized and knew that their religion is true and righteous... and pursued God... and entered beneath the wings of the Divine Presence... [Letter to Obadiah the Convert]
The debate continued. It took four centuries, but the Rambam’s position is now universally accepted.
If you read from the beginning of chapter 26, it is understood that the subject being discussed is primarily about inheritance of the land of Israel. That follows tribal association.
This concept in halacha follows what was decided by the court of Moshe Rabbeinu as recorded in the five books of Moshe. Male converts do not have any tribal affiliation and therefore do not have at present any inheritance in the land. That is the meaning of Rashi’s comment which you cite.
Male converts relate to a higher aspect of the Jewish people which is also alluded to in the same posuk you cite (Devarim 26:11). Namely that they correspond to the innermost aspect of the Jew, also referred to as the Pinteleh Yid or as mentioned in Tanya, the חלק אלקא ממעל ממש.
This aspect corresponds to the whole of the Jewish people (the concept of Tzibbur) which is why, according to some, a male convert is permitted to refer to their connection to the Avot who are the source of the entire nation.
This is also why ultimately male converts will receive an inheritance in the land of Israel which will be donated by all the tribes. The male converts will comprise what is referred to as the 13th tribe.
This concept of 13 also relates to the 13 aspects of mercy, and the level of teshuva connected with Ne’ilah of Yom Kippur, meaning conversion in an inclusive sense.