Way too friendly ex

My bf and I have been together for 8 months. There is one ex gf of his that I dont like. Im calling her Kay. She has sent him three separate messages that left me feeling suspicious of her behavior. She is already in a relationship with a great guy (Ive met) but I think she might be hung up on my bf even though they broke up a long time ago. The first message she sent said merry Christmas. You are like family to me. Im going to take a step back and allow you to be happy- thank you for the gift of friendship. You have practically been the main guy in my life. I assumed and he told me later on that my bf told Kay he was dating me. I met her at a party. After the party- she sent him this message I didnt know how I would feel but if you like her, I like her. My stance on love is *insert bible verse here* you deserve to be happy. love you. That really made my head spin because he never responded to that text. Then - the most recent message (on Valentines day) said Im really glad you are in my life. You are important to me. I dont know what I would do without you. Now - he has reassured me time and time again that he doesnt view her in a romantic way. He has also reassured me several times that the two of them did receive closure and that he was never intending on getting back with her. I dont know if Im in the wrong for staying, if he is in the wrong for staying friends or does she have some hidden motive I need to be aware of?