There is an approach to language called Speech Act theory. It says that the 'meaning' or language is the social functions that words perform, not the semantic mapping of words to say entities in a dictionary or even the more abstract mapping on to intentions hidden in a human mind.
For example when a preacher says 'I pronounce you Man and Wife' he is now meaning anything, his words are doing. Or if I say 'pass the salt' I am not meaning anything, I am instructing. Speech Act says that all language functions like this.
'Keep in touch' is a standard idiom in spoken modern American English (never heard it in UK over 10 years) which marks that a separation is between people who are culturally suppose to have some kind of connection. 'Don't be a stranger' is another phrasing of this.
By saying it upon parting I am indicating that a dialogue or period of dialogues between myself and others have ended. By using these set of words I am indicating that the appropriate reverence for the individuals established by our social relationship to each other. I am also establishing that the period of intense discourse has ended and a new period of less or no contact has started.
Keep in touch closes a period of high discourse and social interaction between people related in our culture.
So the saying of 'Keep in Touch' is essentially a social form. People say it because it would be odd not to say it. If a mother was speaking to a son she almost never sees it would be odd if she didn't say something like this upon leaving.
But if the person who says it will be happy if you call them a week after is a different issue entirely. Actually the use of the term 'Keep in touch' is so scripted by social convention it is just about meaningless. An analogy is let us say you like a girl who you know to be a bit of a Jesus fan. You sneeze and she says 'bless you' only an idiot would wonder if this indicated that she was blessing you and thus valued you and had romantic feelings for you. Bless you is just something many people say when someone sneezes near them in North America; again in 10 years in the UK I have almost never heard it.
The key test would be the response to a call made after the parting when 'keep in touch was made', through a number of verbal indications the person might make it clear if they are happy to really keep in touch, or if they just said it out of social form.