No, the semicolon does not function like the em-dash in this case.
The semicolon is usually placed between independent clauses, or clauses that could function as standalone sentences (Punctuation Guide):
The upperclassmen are permitted off-campus lunch; the underclassmen must remain on campus.
In your example, Cooper is not an independent clause, so a semicolon would be infelicitous. Cooper could be considered nonessential information, further identifying "the big and funny dog." Instead, em-dashes usually stand in for commas, parentheses, or colons. In that case, one could use a comma instead (Punctuation Guide):
I love the big and funny dog, Cooper.
Another alternative would be a colon, which is occasionally used to add emphasis to the last word or phrase of a sentence (Punctuation Guide):
I love the big and funny dog: Cooper.
Finally, you could use parentheses, which are another way to set off nonessential or additional information (Punctuation Guide):
I love the big and funny dog (Cooper).