If every vibration in a wave is replaced dozens of times, is it the same wave?

Depends on the definition of "same" for the cell. It's not straightforward at all.

Most symbols in language where you try this refer to macrostates, collections of microstates that conform to the relevant macrostate pattern. Assigning identity in the first place is highly dependent on what purpose/function you're assigning identity for: what are you trying to achieve by labeling a macrostate as such?

The Ship of Theseus is a classic example in this regard, but complicated by the 'of Theseus' bit: Are we trying to determine ownership? Who built it? Who is standing on it? Those questions lead to very different answers for the thought experiment; they depend on how you define "The Ship of Theseus". Sidenote: examples that include consciousness such as teleportation thought experiments complicate matters even more.

The Sorites paradox / paradox of the heap is one of the simplest thought experiments in this category that mostly avoids that but still runs into the "well, what are you defining a heap for?" issue. One way out is fuzzy membership: Unless you have to act on whether something is a heap or not it is also fine to say "this is 80% like a heap", but as soon as you do have to act on it the 80% membership doesn't cut it.

We don't really have to act on heaps in a way where a sloppy assignment of membership matters, but there are similar things where it matters a lot. A relevant example would be abortion: half-aborting a baby isn't going to work; you're either going to end up with an alive or a dead baby. So defining a fetus fuzzily as "80% human" doesn't help. You are going to have to define a cutoff to achieve a binary distinction and make a decision. Another sidenote: converting fuzzy input patterns to better defined output patterns or even binary distinctions is kind of what neurons do.

edit, addendum: There are of course many dimensions along which to define cutoffs for abortion. The 'is/is not human' distinction is mainly relevant if you start from the premise 'no innocent humans should be killed intentionally' or something similar, which isn't necessarily a given.

A very compressed form of the way I look at it, but hopefully clear and interesting enough.

I remembered once, in Japan, having been to see the Gold Pavilion Temple in Kyoto and being mildly surprised at quite how well it had weathered the passage of time since it was first built in the fourteenth century. I was told it hadn’t weathered well at all, and had in fact been burnt to the ground twice in this century.

“So it isn’t the original building?” I had asked my Japanese guide.

“But yes, of course it is,” he insisted, rather surprised at my question.

“But it’s burnt down?”

“Yes.”

“Twice.”

“Many times.”

“And rebuilt.”

“Of course. It is an important and historic building.”

“With completely new materials.”

“But of course. It was burnt down.”

“So how can it be the same building?”

“It is always the same building.”

I had to admit to myself that this was in fact a perfectly rational point of view, it merely started from an unexpected premise. The idea of the building, the intention of it, its design, are all immutable and are the essence of the building. The intention of the original builders is what survives. The wood of which the design is constructed decays and is replaced when necessary. To be overly concerned with the original materials, which are merely sentimental souvenirs of the past, is to fail to see the living building itself.

— Douglas Adams, Last Chance to See