Hi S.!
Separation anxiety is very common at this age...My son, now 10, was the exact same way at 2! I would get down to his level and acknowledge his feelings letting him know that I know he misses me when I'm gone and that daddy does too so I'm going to hug daddy AND you. Dr. Sears' books about attachment parenting literally saved my sanity. Made me realize that my son is normal, his behavior is normal and that as long as I let him be my shadow, I acknowledge his feeings and love him like mad, he will grow up to be independent, sweet, empathetic and amazing. Here I am 8 years later to say that this is EXACTLY what happened! If I had tried to separate myself from him, against his wishes...If I had ignored him...he would be confused and could actually have become more dependent. My son and I are still extremely close and he truly is the most empathetic and sweet boy I have ever known. Problem with that is....ALL the girls at school are in-love with him... : ) By the way, with my son (and every child is obviously different), his clinging to me lasted until he was about 4. Then he became more independent and it showed up again at age 8 (which really surprised me but it's very normal).....
Just love him up.....Before you know it, he will be off to college and you will be longing for the days when you had a velcro baby!
Best,
S.