One column LIKE another column

Yes.

select a.foo
     , a.bar
     , a.field1
     , b.qux
     , b.fap
     , b.field2
  from table1 as a
inner
  join table2 as b 
    on a.field1 like '%' || b.field2 || '%'

Please don't use the dreaded, evil "select star." Please always try to write your joins using JOIN syntax, not the older style which lists tables using commas.

Are you surprised to see something other than an equals sign in the ON clause? Joins condition can actually be written with all kinds of comparison operators.