The likelihood is that he no longer keeps thinking about her. All we know for sure from the use of the past tense is that at some time in the past he did it. We know nothing from that sentence about the present because 'kept' is simple past.
Thank you very much for your help, Biffo.
But how could you say it when thinking about her is not an obsession and not a regular habit, because he has his own happy life now. He just likes to think back to the time they had together and these memories keep come back from time to time (for instance when he hears the name of the country where they met) and he just enjoys those memories.