Is there any way to find how old one's memory is?
One can find the age of stones, plants, animals, etc. So why not find the age of a memory stored inside one's brain?
We are only barely beginning to understand how the brain works, including memory. We do know that it is a very complex thing. There are many different kinds of memory; many different processes are involved in creating and recalling memories. One factor that plays into a naïve concept of "how old a memory is" is that "remembering" does not seem to be a passive activity; instead, every time we remember a memory we re-create it.
Having said that I don't see a theoretical reason why there wouldn't be a way to determine the age of a memory... but we currently don't know it.
Consider that we don't magically know the ages of stones, plants, or animals. We learned to figure it out from studying those things, seeing how they change over time, and deducing ways to use that knowledge to guess their age. For example, people discovered that in many climates trees alternate a growing season and a dormant season, and this results in the trunk having "tree rings" for each year they've lived. They've also seen that animals have specific life cycles and look different (if only in size) depending on their ages, and they can use that to roughly estimate an animal's age. Obviously this depends completely on the animal in question; different animals grow in different ways and we can tell their ages in correspondingly different ways, and not always to great precision.
For most of human history we were completely unable to tell the age of stones. Only the discovery of radioactivity, and that the atoms inside stones transmute into other elements over the millennia and that this allows us to tell how long it has been since the stone crystallized (i.e. the atoms got stuck where they are), enabled radiometric dating.
Similarly with memory, it is completely plausible that once we understand very well what a memory is, what its exact correlates in the brain are, and how those correlates change over time, it will be possible to tell how old a memory is. But we aren't there yet (see some articles at the end for where we are).
And while it's plausible that we'll be able to know the age of a memory once we understand it well enough, it is by no means inevitable. To give a counter-example, there is absolutely no way to tell the age of, say, a water molecule. We understand water molecules extremely well. They form and can be destroyed, so for every water molecule there is a certain amount of time since it formed, i.e. it has an age. But a water molecule is just an oxygen atom linked to two hydrogen atoms; its properties are dictated by chemistry and don't depend on how long it has existed. There is no way to tell the age of a given water molecule just from looking at it. Similarly, computers give a timestamp to a file when they create it, but if they didn't do that I don't think we could tell how long a file had been on the disk just from examining the disk itself. So it is certainly theoretically possible that once we understand how memory works, we'll find that we can't, in fact, know the age of a memory.
Here are some papers on memory to give an idea of what neurobiologists are looking at these days:
The Regulation of Transcription in Memory Consolidation
This paper looks at the role of DNA transcription in the consolidation of long-term memory (as such consolidation apparently requires new proteins being created)
Structural Components of Synaptic Plasticity and Memory Consolidation
This paper (more or less the same authors) looks at the structural factors involved in consolidating memories by strenghtening synapses and creating new ones.
The many faces of working memory and short-term storage
This recent (2017) paper discusses "Working memory" and how the field doesn't have a single, coherent definition of that concept yet.
Functional neuroimaging studies of encoding, priming, and explicit memory retrieval
This 1998 imaging study looks at what regions of the brain light up in various memory-related tasks, and suggests such imaging studies can be useful to investigate brain function. They were right, but notice how general the areas they see are, and how they only can guess at what might be going on there.
The Molecular Biology of Memory Storage:
A Dialog Between Genes and Synapses
This is a very long lecture from 2000 from one of the authors of the above papers, summarizing their life's research on the molecular and synapse-level mechanisms of memory. These paragraphs give a historical overview of how scientists have thought memory works:
In his Croonian Lecture to the Royal Society of 1894, Santiago Ramo´n y Cajal proposed a theory of memory storage: memory is stored in the growth of new connections. This prescient idea was neglected in good part for half a century as students of learning fought over newer competing ideas. First, Karl Lashley, Ross Adey, Wolfgang Köhler, and a number of Gestalt psychologists proposed that learning leads to changes in electric fields or chemical gradients, which they postulated surround neuronal populations and are produced by the aggregate activity of cells recruited by the learning process. Second, Alexander Forbes and Lorente de No proposed that memory is stored dynamically by a self-reexciting chain of neurons. This idea was later championed by Donald Hebb as a mechanism for short-term memory. Finally, Holger Hyden proposed that learning led to changes in the base composition of DNA or RNA. Even though there was much discussion about the merits of each of these ideas, there was no direct evidence to support any of them [reviewed in 17].
We were now in a position to address these alternative ideas by confronting directly the question of how learning can occur in a circuit with fixed neuronal elements. Kupfermann, Castellucci, Carew, Hawkins, and I examined the neural circuit of the gill-withdrawal reflex while the animal underwent sensitization or habituation, a form of learning in which the animal learns to ignore an innocuous stimulus to siphon when given with monotonous repetition. (We later also extended these studies to an examination of classical conditioning .) Our studies provided clear evidence for Cajal’s idea: learning results from changes in the strength of the synaptic connections between precisely interconnected cells [6, 7]. Thus, while the organism’s developmental program assures that the connections between cells are invariant, it does not specify their precise strength. Rather, experience alters the strength and effectiveness of these pre-existing chemical connections. Seen in the perspective of these three forms of learning, synaptic plasticity emerged as a fundamental mechanism for information storage by the nervous system, a mechanism that is built into the very molecular architecture of chemical synapses 
Notice how twenty years ago, this person was saying we can finally begin to answer century-old questions of what form memory storage takes in the brain to begin with.
A Putative Biochemical Engram of Long-Term Memory
Exciting 2016 article: they've maybe found one place where one memory might be stored! First line of the abstract:
How a transient experience creates an enduring yet dynamic memory remains an unresolved issue in studies of memory.
Same idea in the first line of the abstract of this 2016 article on associative memory:
Neural ensemble dynamics underlying a long-term associative memory https://www.nature.com/articles/nature21682
The brain’s ability to associate different stimuli is vital for long-term memory, but how neural ensembles encode associative memories is unknown.