I'm a bit confused why jisho.org marks the word 直感的 as a noun in a sentence (直感的な人) while it's a Na-adjective in the dictionary. Based on the meaning and translation (intuitive person) it's also supposed to be an adjective. Is there a rule I don't know about or is it incorrectly marked as a noun?
The adjective you asked about is formed from the noun 直感｛ちょっかん｝ in the standard way of adding the 的{てき} suffix. Adjectives formed this way are all な-adjectives.
In the example you give (直感的な人), indeed, it seems to figure as a な-adjective qualifying 人.
You can also form an adverb from it in the standard way of adding に, as for all な-adjectives. Take for instance this sentence (from Tatoeba): 直感的{ちょっかんてき｝には、基本的な意味の最小単位は単語です。
Most な-adjective can also be used as nouns. However with adjectives formed using 的、usually you go back to the corresponding noun by simply removing 的.
For example: あなたは直感を信じないのですか？ "Do you believe in intuition?" There would be no need to add 的 here.
[NOUN]+ 的, such as 印象的, 合理的, etc, but not for "regular" -na adjectives like 綺麗.