Carrier
001 - 大石榴 (big pomegranate)
"Pomegranate" is phonetically identical to the number "16", Liaoning's pennant.
002 - 大葱 (green onion)
Green onion is a stereotypical food of Shandong, the ship's namesake province.
002 was also known as "皮皮虾" or "mantis shrimp" for a while because there was an online poll following its launch before commissioning where users were encouraged to suggest and vote for a name, and "mantis shrimp" came first. The name went viral and caused enough of a stir that Ministry of Defence spokesperson specifically addressed it and said it did not conform to navy regulation for ship names.
LHD and LPD
071 - 雾灯 (fog light)
Some shorten 船坞登陆舰 (landing platform/dock) to 坞登, which is nearly phonetically identical to fog light.
075 - 牛夫人 (madam bull)
Princess Iron Fan, wife of Bull Demon King is a character from the novel Journey to the West. In the film A Chinese Odyssey Part Two: Cinderella there's
where Princess Iron Fan complains to Sun Wukong (who she was having an affair with) that "You use to call me "sweetie" when we viewed the moon together at night, but now that you have someone else you fancy you address me as 'Madam Bull'"
Thus the Madam Bull meme refers to something which was once in vogue but has since been demoted and replaced by something newer. The fast pace of PLAN ship construction in recent years resulted in many Madam Bulls but 075 is the most egregious case because between the launch of the first 075 and its commission carrier 003 began construction and became the most heavily watched ship. Thus 075 became Madam Bull before it was even commissioned.
Destroyer
055 - 小火车 (little locomotive)
"5" in Chinese is pronounced "wu", so two fives together is "wuwu" which sounds like a steam locomotive's whistle
052D - 武二弟 (Wu the Second)
Wu Song is a character from the book Water Margin, younger brother to Wu the Older. Each of the character is phonetically identical to "five" "two" and the letter D respectively.
Frigate
054A - 新青年 (new youth)
Nicknamed as such due to 054A's relative small size and being one of the first PLAN with a clean stealthy hull design.
056 - 小动物 (little animal)
"Animal" in Chinese is phonetically identical to "05" (洞五)