The H-sound in he is the same as the CH in ich. When consonants made with the back of the tongue and mouth are followed by frontal vowels (like /i/ in "he"), they often become palatalized. Unvoiced palatal sounds are made with [ç], as in "ich" and voiced palatal sounds are made with [ʝ], which is the voiced equivalent. Note though that these sounds are phonetic, not phonemic.
/ç/ and /ʃ/ (SH as in she) are undoubtedly similar sounding. That is why many English speakers have trouble hearing the difference between "ish" and "ich". It's no surprise that it's difficult to hear the different between he and she. They are words you are not entirely familiar with.
Keep in mind though, that luckily, or perhaps unluckily, depending on how you look at it, native speakers almost invariably drop the H's on pronouns in connected speech.
So /hçi/ comes out as /i/ (he -> 'e), and /ʃi/ doesn't get reduced (she -> she).
This happens with all pronouns that have an H consonant:
him -> 'im (hɪm -> ɪm)
his -> 'is (hɪz -> ɪz)
her -> (hɚ -> ɚ)
hers -> (hɚz -> ɚz)
And also the pronoun them becomes reduced. In fact, at native speed, you'll likely find that the real pronouns you can't tell apart are him and them, not he and she:
them -> 'em (ðɛm -> əm)