Does self awareness require awareness of one's own self awareness?

No. Don’t be distracted by reflections of reflections!

Self awareness requires only the ability to be honest about yourself. You have to be able to recognize who you are, how you act, what you believe, etc honestly. Most people look at themselves through filters.

Is it self aware to be self aware?
– Yes

Or is self awareness a higher meaning?
– Higher in what sense?
– It is possible to be aware of being aware of oneself. In fact, that can be a useful meditation. Remove all distractions. Focus on yourself and your experience. Then pay attention to what it is that is having that experience. Then pay attention to what it is that is paying attention. Then try to pay attention to what that is. Etc.

We may be aware of all our thoughts and feelings, and can reflect on them. But is true self awareness reflecting on our own existence?
– Not entirely.

Being aware of our purpose?
– Not entirely.

Is that even possible?
– Well who says what “true self awareness” is?
– Who says what your purpose is?
– It is certainly possible, and can be very helpful, to have awareness of many thoughts and feelings, and to reflect on them.
– It is certainly possible to have a sense of purpose, and to reflect on how authentic and important it is to you, and that’s also often very helpful.
– Worrying about what to label “true self awareness” may or may not be useful.
– I find it’s useful to reflect on how accurate my ideas and sense of self are, and to check how authentic my ideas about purpose, and my senses/ideas about how I want to be and what to do, are.
– I find it can be not so useful to quest for a sense of “truth” about one supposed true purpose, or sense of self. I think asking whether a notion or sense seems “authentic” (or maybe “how authentic is it?” is more useful to me.

No, self-awareness does not inherently require awareness of one’s own self-awareness.

It simply means being aware of your own thoughts, feelings, and behaviors, without necessarily reflecting on the act of being aware itself.