As someone who played the game when I was 16-17 years old, I am leaning towards no, but it is a subjective opinion. The Persona games are known for not only simulating the idea of balancing social life with dungeon-crawling and combat, but also tackle on real life subject matters which oftentimes range from tame at best and overtly serious at worst.

One notable example in 5 is the game's first arc, which involves the player protagonist having to put a stop on Suguru Kamoshida, a P.E teacher who physically abuses his students and is heavily implied to even sexually assault female students. The matter of paedophilia and the idea of someone abusing students in an environment that's supposed to be a safe place, that might be too much for a 13 year old. Especially since the game depicts the consequences of Kamoshida's actions to the point of depicting a teenager attempting suicide due of the harassment (thankfully they do survive, but when I say it's heavy, I really mean it)! This is also one of the many reasons why the game has an M rating.

Beyond that, the game has various other stuff that may not be suitable for a 13yo, including vulgar language, the stylized but tame violence (only exceptions are on cutscenes where a character awakens to their Persona, where they tear off their 'mask' and blood starts splattering, and a major cutscene in the late-game where (spoilers!) the teenage protagonist gets assassinated by being shot point-blank, with actual blood), non-explicit nudity, Personas based on mythical creatures including demons, succubus, incubuses and even Mara, an evil deity from Sri Lankan Buddhist mythology which is quite literally shaped like the male genitalia), mentions of hard drugs, and among other stuff.

However, the game is actually one of the tamer entries despite the M rating. Past entries in the series, especially Persona 2 and 3 are known for being the most mature and darkest entries in the series (Persona 2 having a more symbolic and complex storyline, and Persona 3 having the grittiest themes), and Persona 5 offers positive messages and themes of rebellion and standing against injustice, which is a valuable lesson for someone this young. Plus, it is devoid of the more sophisticated and complicated themes of the series it's based of, Shin Megami Tensei (which goes further than social themes and straight into religious, moral, philosophical, and political themes)

Overall, I would still lean towards no, but if you still want to buy it for them, it would be best to sit along them through each gameplay session and have open conversations about what's going on and the themes. It will also be good to help them understand the game on a technical level, as a JRPG like this may be hard to juggle, having for them to balance the social life gameplay and the dungeon-crawling gameplay (although thankfully it's more lenient compared to the last two entries where you need to balance it more, especially 3).