A mobile phone is a traceable gadget that can trace back to you always. No mobile in this world can make you truly anonymous.
If you follow some precautions, you can make it less traceable and maintain your #privacy.
The moment a mobile phone connects to the mobile network to make a call, it sends your IMEI number to the service provider.
IMEI numbers are assigned to mobile phones by the OEMs when they manufacture them. They are hardware coded and can't be changed unless you have a unique hardware box that can flash and alter mobile chip level software.
#Privacy is different for everyone. Not everyone needs privacy in the same way as another.
A Journalist or a whistle-blower working in a repressive regime needs more stringent protection than someone who sits on the couch and watches Netflix every day.
Having privacy is essential for anyone, but what level of privacy you need depends on your threat model and personal preference.
Know your threats:
First, you need to understand what is your need for privacy. Some want privacy from spying apps. Some from their police or government and some from their spouse or stalkers online.
A lot of people have something to protect, and you should build your #threat model around that.