I learn new things every day, which on use becomes knowledge and wisdom over the years. From Buddhism point of view, I can reincarnate in different world and time , can remember my previous birth. What does Buddhism say of samsaric knowledge gained on previous birth? Is it accessible to new reincarnate being?
Habits practiced over many lifetimes persist. Say there is monkey born as a monkey over 500 lifetimes and subsequently born human. The tendency to be a bit jumpy will persist.
The other knowledge you cannot access unless you develop the ability to recall past life. This is not easy and developing these kind of abilities are not connected to the Buddhist goal hence not encouraged.
Without such ability and an average person so cannot access past life knowledge you will most likely not be able to do in this and future lives also.
According to the Tibetan tradition, some children can remember fragments of their previous life, based on the tradition preserved in the Pali Canon that one can intentionally influence one's future rebirths. These children are tulkus, intentional rebirths of somewhat advanced beings (not necessarily "masters" as commonly asserted). Moreover, the Buddha remembered all of his previous lives, which is paradoxical since samsara is beginningless, as noted by Dr. Richard Gombrich. Therefore, the karmic continuum, mindstream or santana, must include memory. However, this is not normally accessible and is usually "overwritten" early in life due to the greater immediacy and intensity of current experience. Dr. Ian Stevenson has documented this is to some extent.
One perspective not mentioned in other answers is that going through the bardo is a traumatic experience. This is the primary reason for not maintaining memory between rebirths. The other primary reason being that memories are a physical aspect of the brain.
That being said, karmic seeds are said to grow, not just naturally, but also because of psychological propensity to repeat the same actions.
I am not sure if it is said that these propensities carry from life to life, or just the karma from them. It seems that it in some way must. Not identity though... but a type of personna or personality. However now I get off topic, so I will cease my ramblings.