This answer will not provide you with any deeper meaning on what it would mean to own your personal spoon and/or fork, but it will tell you how Makise got her spoon and why she wanted a matching fork.
In Steins;Gate: Aishin Meizu no Babel (The story from Makise Kurisu's POV) you can see how Makise gets her personal spoon for her tenth birthday from her father and was supposed to get her personal fork from him a year after that on her eleventh birthday.
Then on her eleventh birthday, she gave her father proof that time machines should be impossible to build. He became outraged over this and instead of giving her anything, he told her he would prove her wrong and erase her from this earth.
So I guess her wanting her personal fork, would mean that she wanted to patch things up with her father and become wanted again by him. Which can be seen when Ferris lets her listen to the tape both their fathers had made 16 years earlier, when she called him after all these years.
Not much had changed though, he still hadn't been able to built the time machine and he still wanted to erase her existence in 2010.