Before answering your question, I did want to address that your foundational assumption is a bit flawed.

Under the Magic Tournament rules, you do not have to name a specific card in order to use an effect like Nevermore. What you must do is provide enough specificity to clearly identify the card you are trying to name. Context and intent is a factor in that decision, as are common colloquialisms. If Both "Approach" and "Second Sun", as phrases, might appear on multiple cards, but as a colloquialism in Magic, they only ever refer to Approach of the Second Sun. Accordingly, the usage of either phrase clearly telegraphs the intent of the user as to what specific card they wanted to hit with a Nevermore style effect.

Thus, under the tournament rules, you would pretty certainly be in the wrong when you used your "Gotcha! I have a similarly named card!" The other player provided more than enough specificity to identify their intent to hit Approach of the Second Sun, and therefore that intent and identification should be considered valid.

Designing a deck specifically for a "gotcha" moment very well might violate the social contract of playing games with friends, so buyer beware before doing something that is fairly certain to be frowned upon by your friends.

Regarding your actual question, multiple copies will trigger independently and give you an additional beginning phase per trigger resolution.