"... people... who can laugh at the same things as we laugh at".
Here's why "as" is correct:
- Making comparisons using the adjective "same":
"They're the same rules as in French."
"Let's meet at the same time as yesterday."
- "same" followed by the pronoun that":
Now, let's look at the following examples:
"This is the same car that we rented last summer."
"Is this the same book that you spoke to me about, the other day?"
In the above examples, the speaker isn't making any comparison. On the other hand, a reference is being made by using "same" (adj), followed by "that" (pronoun), which does its job by referring back to the noun "car" in the first sentence, and "book" in the other.
For a learner, keen on making some personal conclusions about the complexities of a language, this may be confusing. But, with time, one can master such nuances on the usage of certain expressions, I believe.