This is a very shallow and nonsensical statement commonly reproduced as a meme for many different subjects.
"Is Y the new X" is just someone shouting something out loud because of personal frustration or inabilities. Every time.
It assumes X is inherently "bad" (because it is old, because someone else said so).
It assumes Y was supposed to be inherently better (because it is new, because someone else said so).
Waterfall encompass a series of traditional project management techniques and internal corporation bureaucracies. The techniques, by themselves, are not bad. Internal corporation bureaucracies are sometimes cumbersome, an unnecessary hassle but they are also sometimes necessary because companies have to deal with liabilities.
Agile encompass a series of techniques to try to expose internal bureaucracies and other failures as soon as possible, giving the possibility to fix those problems before it is too late. In general, the concept is to manage risk by exposing and dealing with mistakes early on, and as a consequence, increase the general quality of the management and product.
One does not substitute the other 100% of the time and sometimes, the aforementioned "internal corporate bureaucracies" do make it difficult to practice the Agile techniques. And thus, it makes it feel like Agile itself is failing.
Sometimes we do need some chain of events to happen in a sequencial order. Sometimes we can make development more parallel or we can start a simpler "version 1", deploy, and then refine it in a "version 2" after we have real world data to reason with.
Both ways can be done through traditional "waterfall" techniques and Agile techniques. The techniques, by themselves, do not guarantee anything. Agile, for example, must expose problems. And the stakeholders must deal with those problems. Sometimes it means letting of one stakeholder that is actually holding things back more than helping out. But this is something that some stakeholders are not comfortable doing. But by acknowledging a problem and not dealing with it, they are postponing one problem resolution now, for a more expensive problem resolution later. That's all there is.
Hence, when one says "Agile is the new Waterfall" they are only stating that problems that arose when projects were managed in older traditional techniques are still repeating themselves even through the usage of Agile techniques. It's not to conclude that Waterfall or Agile are at fault, it's just to pinpoint that the team/stakeholders are not willing to change and implement the necessarily resolutions to avoid problems.
So, no Agile is not the same as Waterfall, they are just different set of techniques. The teams using them are at fault of neglecting resolving the exposed problems or getting deluded in thinking that there is a Silver Bullet that will magically solve those problems in their behalf.