I'm looking for a 1 word adjective to describe someone who very easily builds close and personal relationships with other people, in a way that almost seems magical to everyone else. Charismatic is close, but not quite, as this person doesn't necessarily inspire devotion or admiration, just quickly becomes people's best friends. The closest word I've found is "palsy-walsy", meaning someone friendly in an intimate/hearty way. It doesn't quite describe the effect it has on the people around them though.
Amiable charisma can be used in the context provided by you in your explanation.
friendly, sociable, and congenial. Example: "an amiable host."
A special magnetic charm or appeal. Example: "the charisma of a popular actor."
"Amiable charisma" is used in various books to describe a person who has friendly disposition with magnetic charm.
1) THE COMMON CASE OF DAMIAN VONGCIR By Uyi Eguavoen.
When Damian first came to Landmark, it was Macaulay Etiebet who was so often talked about, and it was his amiable charisma, his stylish sophistication, and appealing elegance that got the most mentions.
2) 9 Sisters Cursed Since Birth: A Corrupt Family Drowning in Their Deadly Sins By Na 'Veah Rose
I describe Lannah as a charming person, but she can be a victimizer as well. Meaning -- she has the potential of intriguing and enticing people with her amiable charisma.
3) Born Under Punches By Martyn Waites
It was clear to Larkin, even in the short time he had been there, that Tony Woodhouse was popular with both staff and clients. He had an easy, amiable charisma people responded to.
I would suggest the word magnetic:
possessing an extraordinary power or ability to attract; a magnetic personality
The synonyms for “magnetic” cited here are also a fantastic collection to choose from to describe a person who easily draws others in and manages to connect with them in a deep and meaningful way.
alluring, appealing, attractive, bewitching, captivating, charismatic, charming, elfin, enchanting, engaging, entrancing, fascinating, fetching, glamorous (also glamourous), luring, seductive
Update (8/7/2018): Another suggestion could be gregarious, which can indicate someone who easily converses with others and is comfortable to be around (see https://www.merriam-webster.com/dictionary/gregarious, especially "b"):
a : tending to associate with others of one's kind : social gregarious animals
b : marked by or indicating a liking for companionship : sociable is friendly, outgoing, and gregarious
c : of or relating to a social group