1. What is the difference between "My" Tokens and custom object?
2. Can I use "My" Tokens as replacement to custom object?
Tokens and Custom Objects are two totally different and incomparable things.
Objects are the set of related fields having the data. Like Lead, contact, Opportunity, these are the standard objects. Now if you create an object to fulfill your any requirement that will be your custom object.
Tokens are used to print values from the standard object, like if you want to print the lead first name then you will use the Token.
My Token are the custom tokens which you creates to print some values that are not stored in any standard object like in a webinar type program. Consider that I have 4-5 emails, all the email have the webinar date in content, so instead of writing the date in each and every email, we can create a my.token and use that my.token in all the emails. So whenever I will change the date in Token it will automatically get updated in all the emails.