Mars New Zealand lawyer Earl Grey said the Champ tagline was not a registered trademark and Heinz Watties had never applied to register it.
The phrase was simply descriptive of the product and was a relatively insignificant part of its packaging, he said.
Consumers would instead recognise the product by its trademarked logo, which dominated the dog food's packaging.
"Consumers pay more attention to the distinctive and dominant elements of a sign," Mr Grey said.
The trademark Mars was seeking was distinctive because of its unusual combination of "advance" and "advanced", Mr Grey said.
Heinz Watties lawyer Nigel Robb said although its phrase was not registered as a trademark, it had been used on Champ packaging since 1998 and was a recognisable part of the Champ trademark.
"We are here considering what the public will perceive," he said.
"It is inevitable that members of the public who have seen this product on sale for 10 years or longer will make a connection with that tagline ... it is part of the branding."
Mr Robb said there was chance people who knew the Champ tagline could question whether there was a link between the product and Advance.
The appeal was the first High Court case to be heard by former Solicitor-General David Collins, who was appointed as a judge last month.
He reserved his decision.