Yes, that is a (transitive) phrasal verb.
To "run after" someone means to chase them with the intent to catch them and interact with them in some way.
If you parse it as a verb + a prepositional phrase, you would have a meaning of "to run at a later time than someone", which in most cases (including your example) does not make sense.
According to Wikipedia's Phrasal Verbs entry, this qualifies as a "Prepositional phrasal verb".
As for how to recognize a phrasal verb, the best advice I can find (from YourDictionary.com) is that if you read the sentence with its normal, literal meaning and it doesn't quite make sense (or doesn't make sense at all), you are probably dealing with a phrasal verb.