Humans evolved from creatures similar to flatworms, so presumably the mechanism for memory is similar.
It wouldn't surprise me if some of the mechanisms of memory were similar - there are many commonalities in the neurochemistry of creatures of all levels of complexity (that have some form of neural network), and individual cells and organs can 'learn' or habituate in various ways, and there is epigenetic 'memory' where some acquired changes in gene expression can be inherited.
But that doesn't mean we store what we popularly call memories (e.g. declarative - episodic & semantic, and procedural - skills & tasks) in the same way and in the same place(s). In fact, we know from neuroscience that these memories are stored in the brain, and the typical mechanism involves changes in the connectivity between brain cells.
Some scientists like Pim van Lommel or Rupert Sheldrake do not believe memory is stored locally in the brain or body at all, that the brain acts as an antenna for consciousness. Even plants and micro-organisms (like slime molds) that lack nervous systems display goal-directed behavior, even learning, suggestive of a kind of consciousness. Which is why some biologists and physicists see this as suggestive of panpsychism as an explanation for consciousness.
There is no scientific evidence that the brain acts as an antenna for consciousness, and neuroscientific studies indicate the contrary - every recognised aspect of consciousness can be modified, quantitatively or qualitatively, by interfering with brain activity. This would be analogous to interfering with the insides of your TV and finding that you could change the decor of the studio, the plot of the movie, the TV schedule, the teams playing a game, etc.
Not to mention that if consciousness is a transmission it needs a transmitter and some kind of field (force, or particle), to carry the signal, which raises problems of energy conservation, and we know that, while there may be unknown forces or particles, there are no unknown forces or particles of sufficient strength or range to be significant for everyday human activities because we would already have found them - we know what the brain is made of and we know what forces or particles affect it - basically, the choice is between gravity, the very weak 'dumb' force, and electromagnetism, which can barely penetrate the skull and which is easily detectable.
So, no.
It's true that goal-directed behaviour and learning is common in creatures without brains, and some people might consider this suggestive of consciousness - but
all living creatures respond to their environment, and the motile ones do so strategically (e.g. move towards food or light). Simple nervous systems may be involved in multicellular creatures, but much of this can be done mechanically or via chemical cascades without neurons, and much of it is reasonably well understood already, e.g. plant tropisms. Some behaviours can be quite complex and involve learning (storing information), but that doesn't imply consciousness.
We project concepts like 'goal-directed' and 'aware' onto things in the world, and they can have meaning at many levels from simple reflexes & tropisms to human consciousness. We are particularly susceptible to what Dennett calls the 'Intentional stance', where we interpret the behaviour of things in terms of mental properties and agency - we sometimes do this for inanimate objects like machines, sometimes jokingly, sometimes unwittingly.
