I have tried to put your question in the context of modern memory research, but without involving specific models (which would make your question far too broad to answer). Hopefully, this will clarify what purpose the construct of long-term memory serves in modern memory theory, and what exactly is meant by it being "unlimited," without giving a chapter-length answer.
What is long-term memory to memory researchers?
Operationally speaking, long-term memory is memory that fails to exhibit the primary effects that characterize, and which we use to define and study, shorter term and working memory, especially capacity limitations and interference. In a sentence, long term memory is memory that has survived the forgetting process for sufficiently long that the expected chance of forgetting asymptotically approaches zero under a particular model. From the article in the question, some very useful context precedes the given quote:
Closely linked with the idea of memory exercising, is the question of
whether long-term memory has a limited storage capacity or is limitless.
This question is illustrated by the textbook anecdote of the professor of
ornithology who stopped learning the names of his students because each
time he learned the name of a student he forgot the name of a bird.
The bolded quote lies at the heart of what memory researchers mean when they say that there are no known limits on long-term memory. Within the context of current storage-based memory models, we may say that long-term memory is limited if and only if cross-domain interference is known to occur at arbitrarily long time scales (i.e., if new memories can "push out" old memories.)
The existence of cross-domain interference is an oft held anecdotal belief, but there is no empirical evidence to support the idea of any such limit on memories that have 'survived' the consolidation process for sufficiently long. It is particularly helpful to understand the effect known as proactive interference here:
Proactive interference (PI)
Suppose I present you with a list of food items (apple, steak, ham), then test your recall. Then, I present you with another list of food items (orange, carrot, chicken soup) and test your recall again. Your performance will be worse than on the first list, because the first list is proactively interfering with your recall of the second list.
Now, if I present yet another food item list, I can increase the PI effect
until your retention approaches zero. If I instead present you with a
list of car parts, your performance goes back to 'normal'. This is
known as release from proactive interference.
The existence of release from proactive interference effects implies that no or very little cross-domain interference occurs at all, even over relatively short scales, and therefore that no long-term cross-domain interference occurs. Based on this alone, it is fair to say that modern memory science is not aware of any limitation on the amount of information the brain is able to store and retrieve.
I hope my answer has been informative regarding how memory researchers define memory, and what they mean when they say that long-term memory is "unlimited." As I think your question is getting at, long-term memory is certainly not a well defined construct. It isn't something we can observe directly or even indirectly, but tends to be used as a definitional convenience, at least within a research context.
Long-term memory is simply memory that no longer exhibits the characteristic effects that memory researchers operationally define memory by in the first place; a decidedly negative construct. It isn't wrong to say that long-term memory is unlimited, but it isn't particularly useful knowledge unto itself, either.
Note also that, although the distinction between short- and long-term has remained extremely popular since its introduction in the 60's, it is not universally accepted. Erikssonian long-term working memory theory, process memory theory and ecological psychology are all examples of models or schools of thought that either reject or are entirely incommensurate with the idea of a short vs. long-term distinction, or even that memories are "stored."