From your comments in another answer:
Your example sentence (which I moved into your question now):
"The speaker of the poem describes a person
that closely resembles my crush"
You could say:
"The speaker of the poem describes a person that closely resembles the person I am in love with"
If you are in love with someone, you feel romantically or sexually
attracted to them, and they are very important to you. -- Collins
Is there a formal word for "a crush"?
Crush is relatively modern and as you say informal idiom. For less informal phrases you might try:
A person you have a profoundly tender, passionate affection for. A
person you have a feeling of warm personal attachment or deep
affection. A person you love. --Urban Dictionary.
E.g. "She is my love".
Something being so cute that their heart is melting. mostly used by
girls. -- Urban Dictionary.
E.g. "She is my heart".
B1. The cause of someone's feelings: He's the object of my affection. -- Cambridge.
E.g. "She is the object of my affection".
The above is the most formal of the suggestions in this answer, and is more poetic, you wouldn't really use it in every day speech.
In the above example heart is being used figuratively to mean your love, your desire, your attention.
So There's Boy... He Kinda Has My Heart... What does this mean?
It pretty much means you're in love with him, kinda like he has your heart, he's in control of your heart, you gave your heart to him. - Yahoo Answers
e.g. "She has my heart"