Satimā hoti, paramena satinepakkenasamannāgato, cirakatampi cirabhāsitampi saritāanussaritā. (Sekhasuttaṃ)
He is mindful, possessed of the highest mindfulness and discrimination, remembering, recollecting what he has done and said long ago. (Sekhasuttaṃ-SS)
Simply when he is mindful it enhances recalling of memory, if someone says sati has a similar meaning to memory, he is misinterpreting.
This is the phrase that led many people think sati is a form of memory. However, this clearly establishes a relationship between mindfulness and memory, although at the same time it does not just equate the two. Although the standard definition of mindfulness suggests some relationship to episodic memory, the two cannot simply be identified with each other. The reason is that some types of episodic memory can take the form of unintentionally dwelling on memories of the past and related fantasies, which would be the very opposite of being mindful. That being mindful will make one remember better is in fact quite intuitive, and this would apply for working, semantic, as well as episodic types of memory. Being receptively mindful in the present moment, more data related to that moment can be taken in by the mind, thereby furnishing a good foundation for later recalling that moment and its various related details. mindful recognition of the situation in the present moment is indubitably a crucial aspect of successful satipaṭṭhāna practice.( Bhante Analayo, 2016)
When we consider Bhante Analayos explaination, we can conclude that sati (mindfulness) is related to memory but it is not sharing identity with memory.
Mindfulness is not about memory. It simply means bare awareness to the moment. When you are aware of your mind and your object in the meditation dhamma vicaya (Investigation of the nature of reality) helps you identifying the quality of the object. One have to be careful identifying sati and other bojjangas, where they are gathered together.
Bhante Nyanaponika in his book The Power of Mindfulness, explains that the basic, unalloyed form of mindfulness is bare attention. He further explains, the clear and single-minded awareness of what actually happens to us and in us, at the successive moments of perception is called bare attention (Bhante Nyanaponika, 1972).
References:
Anālayo, B. (2016). Early Buddhist mindfulness and memory, the body, and pain. Mindfulness, 7(6), 1271-1280.
Thera, N. (2008). The power of mindfulness. Wheel Publication, 1-75.
(This book published originally in 1972)