When I was thirteen I had a boyfriend for six months. He told me he loved me first. This was also said in Norwegian; I'm not belitteling the words I love you, but they can mean so many things. In Norwegian there's two expressions: "Jeg er glad i deg" and "Jeg elsker deg". The latter is I love you. The first means "I care for you", but is much more common in Norwegian, and used with friends and family far more frequently than "I love you", which is mostly (though not always) reserved for romantic relationships (and ice cream).
Anyway, yes, from personal experience I will say that a 13-year-old is perfectly capable of fancying themselves in love. I too believed I loved him deeply and I remember how outraged we both were when his mother said it was only puppy-love!

Ah, those were the days...