"Mine" is simply the possessive pronoun.
It can be the subject: "Mine is the blue one."
It can be the subject complement: "The blue one is mine."
It can be the direct object: "I gave mine to the boy."
"My" is the possessive adjective.
"My thing is blue."
"That's my thing."
"I gave my thing to the boy."
Listen to any sentence in any language wherein nearly the same sound is in the end and beginning of contiguous words. Even if there's supposed to be a break, the two just flow together. Sometimes the sounds don't even need to be similar, just that the phrase is common enough.
Example in English of the latter: "Did you eat yet?" = "J'eet jet?" In some regions the second j will be replaced with the original y.
and, of the former: "I can't take it!" = "I can'tak'it!"