In spoken language we use shorter or longer pauses to break up what we are saying into chunks that allow the listener to follow along comfortably. If you say I have a dog his name is George as one string of words without a pause, you give the listener little time to process each separate thought.
In written language the pauses are represented by punctuation, but they serve the same purpose. And rightly or wrongly, writers who fail to punctuate consecutive independent clauses, as in the dog example, may distract their readers and not be taken seriously. In any case, taking care with punctuation demonstrates courtesy to the reader.
As to what you were taught about not using a possessive pronoun such as his in a standalone sentence, this may be true if you say out of the blue His name is George. But in normal conversation the context will make it clear who the his is referring to (in grammatical terms, its antecedent).
To conclude, it is certainly advisable to avoid writing two or more consecutive independent clauses with no punctuation. You can do this in three ways:
- By making the second clause into a new sentence
- By separating the two clauses with a semicolon
- By inserting a conjunction between the clauses
The term for two consecutive independent clauses separated by a comma (a fourth possible option) is comma splice. Comma splices are generally also considered run-ons and hence to be avoided. There are however circumstances in which a comma splice is not only acceptable but may also be stylistically the best choice.