Hi Noah,

Very good question. I haven't seen a good answer to this question yet.

In any case, it is important to divide the question into two parts.

The first part is the specific question about Rebbi Yehoshua Ben Karcha, and according to Rashi, who writes in several places that he was the son of Rebbi Akiva. On this I would like to point out the words of Tosfos (Shabbas 150a, Bava Basra 113a, Bechoros 58a) and other Rishonim who do not accept Rashi's opinion. It seems that most of the differing opinions about Rashi are due to many difficulties and problems with his opinion.

In any case, it can be said that perhaps Rebbi Yehoshua did not study with his father, since it is known that his father was not at home for many years (24 years at least), although it is part of the discussion when exactly was Rebbi Yehoshua born, if is actually Rebbi Akiva's son. But this is a hypothesis and I have no authoritative source for this.

The second part refers to the general question about Gezeira Shava which is not studied unless there is a tradition about it, here too there is a lot to expand on what exactly is the requirement that a person should hear from his rabbis, and what is there for him to learn from himself. But for the very question of how it is possible for one Tana to learn Gezeira Shava that he learned from his Rabbis, and the Tana who disagrees with him cannot learn the Gzeira Shava from his Bar Plugta who DID learn it from previous generations, or from his Rabbis, and especially here, that it may even be his father, here too I would like to refer to the words of the Tosfos (Shabbas 97a) that asks the question, and says something interesting. He says that what was known from Har Sinai, is that there are a clear number of Gezeiros Shavos there are in the Torah in general, and as Tana'im argue whether or not to learn something from a Gezeira Shava, they actually argue whether this specific study is included in the general number. In our case, this excuse is a bit far-fetched, after all. This is about Gezeira Shava on a word from the Megillah and not a word from the Torah, so it is a bit difficult to assume that this specific Gezeira Shava was included in the number from Har Sinai, especially since there are no other Gezeiros Shavos learned in the Megillah, so it is difficult to understand what is the proposed replacement for this Gezeira Shava.