“Tortoise” and “taught us”

I’m reading Alice in Wonderland, and found the following dialogue:

“The master was an old Turtle — we used to call him Tortoise—”

“Why did you call him Tortoise, if he wasn’t one?” Alice asked.

“We called him a Tortoise because he taught us.”

What is the relationship between “he taught us” and “Tortoise”? Is this some kind of pun or joke?

The passage is funny because the Mock Turtle acts as an authority figure but uses abnormal logic and reason. This pretense of authority and twisting of logic are ongoing motifs of Alice in Wonderland.

By saying, "we called him Tortoise [ˈtʰɔː təs] because he taught us [ˈtʰɔt əs]", the Mock Turtle claims that this similarity of pronunciation is a valid reason to call a turtle "Tortoise". He implies that his own pronunciation-based logic is more valid than the actual ways to distinguish turtles and tortoises.

Note: In other dialects of English, such as American English, the pronunciation of "tortoise" [ˈtʰɔɹ ɾəs] is quite different from "taught us" [ˈtʰɒ ɾʌs]

This is a pun that needs to be understood in its context. Although he was a Turtle, his pupils called him a Tortoise, because:

'We called him a Tortoise because he taught us!' said the Mock Turtle angrily: 'really you are....'

It's a pun by Caroll.

It seems that Americans don't get this pun, because the American pronunciation of "tortoise" differs from the English pronunciation. Because Carroll was an Englishman and was writing to a British audience, it would have made sense to them.

In British English, "tortoise" is pronounced nearly exactly the same as "taught us", and that's why the students called the Turtle a "tortoise", even if he wasn't.