According to the accepted answer of Determining which US zipcodes map to more than one state or more than one city?, there are only 13 zip codes in two states. Reading down further on the page, though, you can see that back in 1994, there were 153 zip codes that cross state boundaries. That response doesn't actually tell you what the 153 anomalous parties are, though. Another response right above that one from 2016 found and listed 93 multi-state zip codes, but is missing at least one code that I know of.
The more correct answer of 153 is still quite dated, though, so I resubmit the question - how many US Zip codes span state boundaries and what are they?