How are key value pairs stored in salesforce?

To have key value pairs stored in salesforce -

You could create a custom object called e.g. "Code" with these fields:
Code Table Name (could shorten this to just Type)
Abbreviation
Valuewhere all rows from the old system from one code table have the same "Code Table Name" value. Then all the old system data can be kept in one custom object.

A custom metadata type - a special form of custom object - could also be used.

Or even just Apex code could be used if the number of values is relatively small and the values will never change. So overall, think about the number of records, whether they change, and who might need to change them which affects how you store them.