Learn English – A word for “intimate friends” without any kind of romance

"My ex-wife's child" is the proper description for your blood-or-marriage relationship with this child.

If you are back with your ex, you can call the child your step-child. If you adopt them, they are now your son/daughter.

Unlike "Significant other" (SO), there is no terminology yet to name this particular relationship that you can use and expect to be understood exactly.

Having said that, whose business is it that you would need to explain yourself so precisely? If he loves you like a father, and you love him like a son, and the mother goes along with this, describe him as your step-son. If someone questions you, politely steer the conversation the way you want it to go. "As I was saying, it was a great game. Have you been to any games lately?"

"Counterparties" is the word that is used in the financial industry for parties at either end of a transaction. It's most frequently seen as part of the phrase "counterparty risk", which is the risk that a counterparty won't be able to hold up its end of the bargain.