W.C.
Your daughter is still not really in the stage of playing "with" some one. She is playing "next" to someone. So don't expect any great social interaction for over a year or so. Five maybe.
Her behavior is not uncommon for a three year old. Have fun with it. Find things that she can collect one at a time on shopping trips to the grocery store. Then set them up one at a time in her room.
She is in a stage of many stages. She and you will just have to live through this one. As her language skills develop living through her stages will become easier.
You are doing fine, Mom. And so is she.