Lose the quotes around the table name and the column name.
If you MUST use reserved words as identifiers, then use back ticks (just left of the 1 key), not quotes.
INSERT INTO jobs.companyID
SELECT * FROM companies.companyID
WHERE jobs.company=companies.company;
or
INSERT INTO `jobs`.`companyID` SELECT * FROM `companies`.`companyID` WHERE `jobs`.`company`= `companies`.`company`;
but save yourself a LOT of typing (and errors) and don't use reserved words as identifiers.
BTW, your SELECT doesn't make sense.
You seem to be trying to insert into one column (companyID) multiple fieds SELECT * , and you need to SELECT ... FROM a_table_name, not a_table_name.column_name.
So, build the SELECT first, until it SELECTs the results you expect, then add the INSERT part.
Good luck,
Barry.