A simple option is to use instr, which shows whether surname exists in fullname:
SQL> with test (id, fullname, surname) as
2 (select 1, 'John Doe' , 'Doe' from dual union all
3 select 2, 'Peter Parker' , 'Parker' from dual union all
4 select 3, 'Brian Griffin', 'Brian Griffin' from dual
5 )
6 select *
7 from test
8 where instr(fullname, surname) > 0;
ID FULLNAME SURNAME
---------- ------------- -------------
1 John Doe Doe
2 Peter Parker Parker
3 Brian Griffin Brian Griffin
Another option is to use one of UTL_MATCH functions, e.g. Jaro-Winkler similarity which shows how well those strings match:
SQL> with test (id, fullname, surname) as
2 (select 1, 'John Doe' , 'Doe' from dual union all
3 select 2, 'Peter Parker' , 'Parker' from dual union all
4 select 3, 'Brian Griffin', 'Brian Griffin' from dual
5 )
6 select id, fullname, surname,
7 utl_match.jaro_winkler_similarity(fullname, surname) jws
8 from test
9 order by id;
ID FULLNAME SURNAME JWS
---------- ------------- ------------- ----------
1 John Doe Doe 48
2 Peter Parker Parker 62
3 Brian Griffin Brian Griffin 100
SQL>
Feel free to explore other function that package offers.
Also, note that I didn't pay attention to possible letter case differences (e.g. "DOE" vs. "Doe"). If you need that as well, compare e.g. upper(surname) to upper(fullname).
SOUNDEX()is an outdated function that doesn't really have good use in this century.