Do all living humans literally descend from one ape?
Just 6 million years ago, a single female ape had two daughters. One became the ancestor of all chimpanzees, the other is our own grandmother.

Short answer
Whether or not the claim is true actually depends upon the specific DNA sequence you are considering. The claim is true for sequences without recombination (such as mtDNA and most of the Y chromosome). The claim does not hold true for most of the genome and it is therefore misleading.
Long answer
Basic knowledge required
First, if you don't know the concepts of

allele
haplotype
ploidy
selection
genetic drift
segregation
recombination and crossover

, then you might want to have a look at a short introduction of evolutionary biology such as Evo101 for example. The below answer will make more sense to you with some a priori knowledge.
Coalescence
Imagine there is today a population of $N$ individuals. We are going to look at this population backward in time (which is a little bit unintuitive). For simplicity, we will assume asexual reproduction, no recombination, absence of selection (only drift) and constant population size of $N$ individuals. The assumption of no recombination is key here. The other assumptions are details just to make my job at explaining coalescent theory easier.
We start at time t=0 and are going backward in time. If two individuals are siblings at $t=0$, then, it means that in the previous generation, they share the same parent. Therefore, if we connect individual from generation $t=0$ to their respective parents at generation $t=-1$, we see that some group of individuals (the siblings) &quot;coalesce&quot; to fewer individuals (their parents). Individuals at generation $t=-1$ that are not connected to individuals at generation $t=0$ are those who did not reproduce (or who did not leave offspring that have survived until adulthood).
You can keep iterating over generation $-2$, $-3$, $-4$, etc... At each generation, the number of individuals that are connected (that our ancestors) to present day individuals is diminishing as more and more coalescence happen. After a sufficient number of generations, eventually, only one parent will remain. This single individual will be the ancestor of all present individuals.
If you enjoy math, then note that such process is actually a simple a branching process for which it is relatively easy to calculate the expected time (and the whole PDF) to the next coalescent event and the expected time (and the whole PDF) until all individuals in the population coalesce.
So, assuming no recombination, if you consider individuals in a modern population, then you can be sure that all individuals share a common ancestor who is a single individual, at some point in the past.
Incomplete lineage sorting
The process of all individuals coalescing into a single ancestor takes a little bit of time. For recently diverged species (say species A and species B), it is possible that this common ancestor is older than the time of divergence between species. If this is the case, it is then incorrect to think of a mother who gave birth to two individuals, one becoming the ancestor of all individuals of species A and one becoming the ancestor of all individuals of species B. In such a case, we talk about incomplete lineage sorting.
Note that when we talk about 'incomplete lineage sorting', we are generally talking about specific genes as not all genes coalesce the same way when there is recombination and segregation (see below)
Recombination
Now the trick is that in the presence of recombination, different sequences coalesce differentially. It also means that when you look at two sister species, some sequences can show incomplete lineage sorting while others don't.
The presence of recombination therefore makes the claim false!
Do we recombine?
Yes, all of our genome recombines to the exception of two sequences

mitochondrial DNA (mtDNA)
Y chromosome not including the Pseudo-Autosomal Region (PAR)

mtDNA is maternally inherited (without crossover) and transmitted to all offsprings, while the Y chromosome is paternally inherited (without crossover) and transmitted to sons only.
So, is the claim true?
mtDNA
It is true that there was an ape who had two daughters. One daughter's mtDNA became the ancestor of all mtDNA we have in humans today, the other daughter's mtDNA became the ancestor of all mtDNA that exist in chimps populations today.
Of course, this daughter is the common ancestor of all modern mtDNA but is not necessarily the Most Recent Common Ancestor (MRCA). In humans, the MRCA to all modern mtDNA is called Mitochondrial Eve (mt-Eve or mt-MRCA).
Y Chromosome
The same claim is true for the Y chromosome (although I am not sure there was really not a single recombination event happening on the Y chromosome since). The MRCA to today's all Y chromosome is sometimes called Y-chromosomal Adam (Y-MRCA).
The names 'mt-Eve' and 'Y-chromosomal Adam' have of course been chosen in reference to the creation myth of Abrahamic religions.
Rest of the genome
The Y chromosome and mtDNA together represent less than 0.2% of our genome. The rest of our genome does recombine.
The claim does not necessarily hold true for 99.8% of the genome. I would therefore tend to think that the claim is misleading at best, wrong at worst.
Adam and Eve
We can call Adam the MRCA of all Y chromosomes today. Only Y chromosomes, not the rest of the genome. We can call Eve the MRCA of all mtDNA today. Only mtDNA, not the rest of the genome.
Of course, Adam and Eve did not live in the same time period and did not live at the same place. Adam and Eve never met. And of course, there were plenty of other humans around. They are in no way the first two humans.
Other popular source of knowledge
PBS Eons - The Two People We're All Related To