No--canis meī looks like "dog of me," which is not how Latin did possession, for the personal pronouns like "I, you, we, you all."
Those pronouns (in the nominative forms: ego, tū, nōs, vōs ) all have their own special, possessive adjectives:
meus, mea, meum , "my"; tuus, tua, tuum , "your" (when YOU = 1 person); noster, nostra, nostrum , "our"; vester, vestra, vestrum , "your" (when YOU = 2 or more people).
These possessive adjectives will describe the 'item possessed' in the same way that any adjective does:
if magnus canis = "a big (male) dog" and magna canis = "a big (female) dog," then "my dog" is either canis meus or canis mea , depending on the sex of the dog.
A sentence like, "I love my dog," which makes the dog the direct object of the verb, puts "my dog" into the accusative case:
Meum canem amō or Meam canem amō .
If you wanted to say, "The boy's dog," you'd have Canis puerī , where puerī is a possessive ( = genitive case) noun. "The girl's dog": Canis puellae . "The senator's dog": Canis senātōris . (So, I've shown 3 different nouns with 3 different genitive-case forms, because these are 3 different "declensions" of nouns.)
Please ask if you have any more questions, or if anything I've said here hasn't been explained well enough.