Does anyone else experience books, like literally feel the physical pain of characters? What would you call that?
I would call it emphathizing with the characters, feeling empathy for them.
Empathy : the action of understanding, being aware of, being sensitive to, and vicariously experiencing the feelings, thoughts, and experience of another of either the past or present without having the feelings, thoughts, and experience fully communicated in an objectively explicit manner; also : the capacity for this.
Some in the Roman Catholic tradition suffer literal wounds in sympathy with the suffering of Christ. These wounds are called stigmata.
Reported cases of stigmata take various forms. Many show some or all of five Holy Wounds that were, according to the Bible, inflicted on Jesus during his crucifixion: wounds in the wrists and feet, from nails; and in the side, from a lance.
(This is the closest definition of the OP's request to literally experience someone else's pain: Christ's wounds become their wounds.)
Words that are showing strong identification with a character, but not the same literal pain, include:
experienced or realized through imaginative or sympathetic participation in the experience of another. A vicarious thrill
Here, a reader might have great sympathy for the character's pain, but not necessarily experience the pain itself.
the action of understanding, being aware of, being sensitive to, and vicariously experiencing the feelings, thoughts, and experience of another of either the past or present without having the feelings, thoughts, and experience fully communicated in an objectively explicit manner; also : the capacity for this
A reader with great empathy might experience the pain of a character, as empathy includes the notion of vicarious experience. That reader might weep with sorrow at a character's loss or pain. But if a character experiences the pain of a migraine headache, not even the most empathetic reader would seek a Toradol injection.
Say, "you can relate to the characters well"
Relate to someone/something — Macmillan Dictionary
to be able to understand a situation or the way that someone feels and thinks
"Martin had always found it easier to relate to women."
"The programme deals with scientific subjects that ordinary people can relate to."
That's sometimes called "getting under the skin of the characters".
get under someone's skin PHRASE
3 Reach or display a deep understanding of someone
‘I know it's a cliche but I thought the author really managed to get under Clara 's skin in a way which made us empathise with her.’