As others have said, this game has died and been reborn several times. Just thinking back to the gateway of Calypso, Port Atlantis, can show how many times the game has had pretty big changes. Who started by the tree? Who started at the back of the ship? Maybe you remember all the light bridges as vehicles were introduced... or perhaps you started on the newbie island and then came into the port we know today. Maybe you're someone as old as me or older, not even spawning in Port Atlantis. How about death? Did you ever run as a ghost? Did you just teleport to the last terminal used? Or did you start with it taking you to the closest terminal? So much has changed over the years that the game as it sits now is vastly different from how it was in the old days. For many of us that were around back then, this game felt like it captured lightning in a bottle and for many of us, that lightning escaped and we've just stuck around hoping to get it back in. I'm one of those - for me that lightning was the community that the game had back then and for me, I noticed that with each new area people could go, the community thinned a bit more until eventually the world started to feel empty. For me, the death of the game was when the world fell silent and my friends list slowly shrank until now only a few old-timers are left on it, but the game is still alive for others and likely will be for a while.
I've been around for 20 years and seen PE/EU go through several ups and downs - I don't foresee it shutting down any time soon, but I'm also pretty sure it will come to an end one day. It's highly unlikely that it will just shut down suddenly, the game still turns a profit, but as users dwindle and the profit margins get thinner, we'll eventually start seeing the signs. Predicting when or how is something people have done for a very long time. Just a few off the top of my head:
Updating the engine to Morrowind version of Gamebryo, people said it would kill the game.
VU6.0 loot changes, people swore would be the death of the game.
The run speed nerf that killed the ability to kite farm most mobs without any armor, that pissed off a lot of people.
CryEngine update, lots of people kicked and screamed up to the time it deployed because they were on ancient PCs.
Every time the game has gone on - large groups would leave, but others would join with the new features. I think the tell-tale sign will be when there are a series of universally derided changes to the game. As long as there are still new players and the game remains profitable, they'll keep it running. There were times before that could have spelled doom for the game, such as the 2008 financial crisis or back in the very early days when MA was raided on allegations of using unlicensed software. So, I guess there is an answer - what will the end of EU look like? It will look like a situation where the game is no longer profitable enough to justify keeping it running. That will likely take longer than with many games though, as EU has a very deeply entrenched player base that has spent an incredibly sum of money... that alone will keep things rolling.
Now pardon me as I go log in for some nostalgia before disappearing for a while once more.