How to get part of the string that matched with regular expression in Oracle SQL

One way to do it is with REGEXP_REPLACE. You need to define the whole string as a regex pattern and then use just the element you want as the replace string. In this example the ColorID is the third pattern in the entire string

SELECT REGEXP_REPLACE('product=1627;color=45;size=7'
                         , '(.*)(color\=)([^;]+);?(.*)'
                         , '\3') "colorID"  
FROM DUAL;  

It is possible there may be less clunky regex solutions, but this one definitely works. Here's a SQL Fiddle.

Try something like this:

SELECT REGEXP_SUBSTR(REGEXP_SUBSTR('product=1627;color=45;size=7', 'color\=([^;]+);?'), '[[:digit:]]+') "colorID"
FROM DUAL;