zip code shapefile by state

Zip Code Tabulation Areas (ZCTA) are a US Census product derived from the Zipcodes defined by the USPS which, as a federal entity, may cross state and other administrative boundaries. They are semi-fuzzy boundaries based on the logistical delivery of mail and are subject to changes between censuses.

Per the census mandate, census data must adhere to state boundaries, so that is "where they draw the line".

The following is partially speculation on my part, but I believe the state reference is intentionally omitted for reasons of non-coherence with other federal data authorities.

The ZCTA lookup tables you mention should be considered the official index to a FIPS code. As a practical matter, one could also do a spatial join to the containing state/county boundary to determine the state most census ZCTAs are within.