my "short wave" radios (the type you use to listen to music and news in exotic countries) have frequencies ranges roughly between 5 and 20 MHz (with big gaps). A cursory look suggests that there is little / no overlap with typical US HAM frequencies for mine at least. I imagine the bands on the inexpensive SW radios are just those for commercial stations (not the same as for HAM).
HAM is a nickname for Amateur Radio. Amateur radio has very specific legal meaning in the US with very specific bands reserved for transmission by those folks with the related license. The bands used by HAMs are usually intentionally different from those for commercial stations (I imagine). The easier to get HAM license are for frequencies well above those SW radios. It takes a more difficult to get license to get in the SW region.
Of course, you can listen to pretty much any band if you have the appropriate receiver (exceptions for cell phones etc), as it does not take a license for listening. Some wideband digital receivers used by some hams can tune in to just about any frequency over enormously wide ranges (expensive) -see below. But the HAMs can only legally transmit over specific narrow bands. And the HAM transmitters are all likely software or hardware-limited to transmitting only on the specific HAM ranges AFAIK, although some have extended reception.
Basically, commercial SW radios (as in British colonials in Kenya listensing to the BBC over a Scotch) and HAM radios are apples and oranges I believe.
example:
this is a wide range receiver that anybody could buy without license in the US, I think: (from the Yaesu web site)
"Yaesu VR-5000 COMMUNICATIONS RECEIVER
0.1 - 2599.99998 MHz* LSB/USB/CW/AM-N/AM/WAM/FM-N/WFM
All-Mode Wide-Band Receiver (*Cellular blocked)
The exciting world of communications monitoring, from Longwave to Shortwave to Microwave, comes to your home or station with Yaesu's new VR-5000 Communications Receiver.
Professional features, professional ergonomics, and professional DSP-based selectivity* are yours to enjoy today-and only from Yaesu!
snip
Preset Shortwave Broadcast Memory Bank
Featuring a handy world map showing the station’s location, the special Shortwave Broadcast Station Memory Bank includes several different frequencies from a number of popular shortwave stations, including Voice of America, the BCC, Radio Japan, Voice of Russia, etc. The operating frequencies may be changed by the owner, to keep up with changing station schedules!
"
as you can see, it can listen to any frequency (except cell) in that wide range, including SW broadcasting. Kind of like your SW radio on super steroids.
Of course, you cannot legally transmit on all these frequencies without the related license. For that you need a transmitter also and these are normally frequency limited. In some countries you even need the license to buy the transmitter - not in the US.