No. Who is to say that a culinary NPC in this game could not come up with a new meal called a burger (made from pre-existing in game ingredients) and that NPC's family name just happens to be McDonald. It doesn't break immersion because it is possible within the setting and so are these items from this event. SE could easily write such in to the lore and lore itself is always evolving so this event merely forms part of that lore.
Themed weapons...already have such things in this world, themed minions...already have such things, weird mounts...also had such for a long time prior to this event and this is just the latest weird mount. As for the quests they just need to be written within the confines of what could/might exist within the in game world. The anniversary event was more immersion breaking than this by a million miles when you were teleported to the offices of the developers, talking to them about the game and seeing concept art posters for it on the walls plus a developer riding a fake unicorn while playing the game itself on an in game tv screen.
The immersion excuse is a subjective one which holds very little if any water because of how subjective it is but given past events and how much content in this game already could fall into that category, it is a bit late to decide that is not part of what forms this game. If anything the OP is breaking their own immersion by applying real world perspective on what is essentially a character that has no knowledge of our world. From an in game character perspective none of what is being done with this event is impossible. Hell, before our character got off the dang chocobo cart at the start of the game the character probably thought a giant talking crystal was impossible and same with a million other things that has happened to that character.
The OP says if do not agree with her then we are not real FF fans, I could quite easily turn around and say if you are against this event you are not a true FF14 fan because these events are part of the makeup of this game's development and has been for quite some time. I assume the OP would also consider chat between players which often is not game related or in keeping with the ideology of the Warrior of Light and/or events surrounding his/her story is immersion breaking too but I am betting she/her just ignores such chat and likewise the OP should just ignore this event if it is not his/her cup of tea.