Personally, I see many similarities between Si and Fi at their core. Both offer a sort of internal barometer for the user of how their internal state is at any given time. Why are Si people sometimes stuck in their ways and routine? Because they know what feels right for them and why differ from that positive, physical experience? Fi as you know, is a constant meter of someone's emotional state and does much the same. With both functions, you're using your external functions to alter the external environment to ensure that inner state is of equilibrium.
I don't feel they are that similar; not any more than the similarities between all introverted functions. I think Jung's dividing of the introverted types as rational/irrational is insightful for that reason. Fi does not equal a meter for one's emotional state, rather that is
part of the process, and it is not a matter of what feels right, but of what accords with an internal construct of value that has been carefully created and refined. The focus of Fi is building this understanding.
As with all introverted types, there is a comparison of some inner construct and aspects of reality. The main differences can be in the focus of the comparison (thinking --> compare to find what is logical or aligns with one's inner construct of logic; sensing ---> compare to find what IS, aka, what is a consistent aspect of reality across time; feeling ---> compare to find what is meaningful/valuable to the human experience, using the self as a prototype for human; intuition ----> compare to consider what WILL be, aka how reality will likely develop, by ignoring the surface in favor of hidden trends/meanings/symbols that occur consistently across time).
Si types could be called "idealists" as well. They often have a sense of what reality should be like, if it were to be very consistent across all contexts, and these realities suggest particular meanings. A certain action consistently means a particular thing (at least for that individual). Although they also seem to know this sense was built off of their relatively narrow experience (which can be scary for them - to always know there is so much unknown), and this can be what can drive them towards new experience and to revise previously held notions of reality. Sometimes they seem very interested in the experience of others, shown through their tendency to engage in story-telling.
However, the "should" for a Fi is not concerned with consistent realities, but instead aims for there to be particular meanings attached to experience, meanings that they have yet to ever truly see/feel in reality; and so these manifestations of meaning as experience can vary widely. The same thing can have a very different meaning given the context. It is almost the exact opposite of Si in some ways, if you think about it.
I know the word "irrational" is not too flattering, but perceiving suggests not reasoning, but rather "seeing" something as just "being". It is more like "gathering" and "experiencing". Pi types seem to explore and review impressions and sometimes things pop up as insights (?). Introverted Rationals (prefer the word rational to judging - connotations are more accurate) are basically always
constructing, even if that process for a Fi type is more exploratory & less dry/technical, akin to a process like sculpting or sketching. There is a deliberate effort there, even if it seems very open-ended. But Jung notes that the inner worlds of Pi types are
spontaneous and how their impressions seem to arise on their own (they may not connect them with themselves), in contrast with how their exterior can appear (more deliberate and cautious).
Personally, as I've gotten older, I recognize how much of a problem having relatively inferior Si has been for me. People often overrate Si in INPs, no realizing as much of our issues can stem from poor Si. Because of silly ideas of what sensing is, people see artistic creativity, fact/info gathering via research, or enjoyment of "sensory activities" in INPs and think this is Si - nope.