Let's take for instance the word "chérie". I feel like I'm hearing [ʃɪːʁi] instead of [ʃeʁi]. I also speak Spanish and I feel like the French /e/ sound is not as [e] as Spanish has. Am I hearing wrong?
The vowel retranscribed /e/ in Spanish typically isn't a cardinal [e], but a vowel more intermediate between [e] and [ɛ] (which makes sense since Spanish doesn't contrast /e/ and /ɛ/). So while they have the same symbol, /e/ in Spanish and in French doesn't really refer to the same phone range:
Vowel charts of French and Spanish, as reproduced in Teles & Huey 2020 THE EFFECTS OF THE FRENCH VOWEL INVENTORY ON VOWEL PRODUCTION IN SPANISH SPEAKERS
The cardinal vowels of the IPA were codified during the 19th century by mostly French and German scholars, and as a consequence match very closely with the vowel space of conservative speakers of French, except for /ɔ/, as you can tell from the chart above.
So the perception of the OP that French /e/ is produced higher than Spanish /e/ is perfectly sound. The transcription of French /e/ as [ɪː] is a lot more contentious however. [ɪ] is not just intermediate in height between [i] and [e], but also centralised. A vowel that is still fully front but higher than the cardinal [e] is more accurately transcribed as [e̝] (e with a raising diacritic). The length is also a bit puzzling (it'd be very unusual for the first syllable of chérie to be longuer than the second one) and it'd help if the OP would clarify which languages they were using as points of comparison here.
There are French varieties whose /e/ is a bit different than those presented in the chart above. Midi French, like Spanish, doesn't distinguish /e/ and /ɛ/ (with the merged phoneme transcribed as /e/ or /E/ depending on the author) and instead uses a slightly higher than the IPA cardinal [ɛ] in closed syllables and a slightly lower than cardinal [e] in open syllables, but that's the opposite of what the OP is asking about. Belgian French has slightly higher mid vowels than France French in a way that's closer to what the OP is positing (i.e. /ɛ/ is [ɛ̝] and /e/ is [e̝]) but also has vowel length distinctions that'd stop short /e/ from appearing as [e̝ː] (and certainly never [ɪː]).
Additionally, the mid vowels can stray from the cardinal position in the chart above when they're influenced by their environment. In particular, /r/ tends to lower and back adjacent vowels, and the stressed vowel can influence preceding mid vowels in the same word: low vowels like /a/ or /ɛ/ can lower /e/ toward [ɛ] (as in général) while high vowels like /i/ /e/ or /u/ can raise /ɛ/ toward [e] (as in rêver).