On my contacts app it says "me" next to my mom's contact but I'm not entirely sure what that means. Does this mean she's using my number on her phone for imessage or is it simply that I am on the same Apple ID as her?
That means that contact has been designated for Siri to use when she needs personal information about you.
For example, if you have a contact created with your home address you can then ask Siri for "directions to home". She will look up the home address attached to the "me" contact.
You can change which contact the "me" is attached to by going to Settings > General > Siri > My Info and selecting a different contact.
Apparently this setting also can be changed here: Settings > Mail, Contacts, Calendars> My Info.