Are all smart card readers the same?

Smartcard communication protocols follow well-established standards (connector size and location, voltages, signals, logical transport protocol...) so chances are that card readers are indeed interchangeable. If the bank sent you both the card and the reader then there is a slight probability that they did something fancy which compromises interoperability, but I deem this probability quite low.

(Personal experience with people in the smartcard industry has taught me that these standards are considered by them as part of some kind of revealed religion; they will not trespass without a powerful reason.)

What you should look for is CCID (this is a USB standard for smart card readers). While the card<->reader interface is standard, the way the reader talks to the host PC has long been a wild west. Unfortunately you can still buy hardware that has either fancy SoC connections (IIRC some laptop embedded Ricoch and/or Broadcom devices where such?).

Some manufacturers tweak/break CCID as well, so your best bet is this list:

https://ccid.apdu.fr/ccid/section.html

Avoid everything in the red list.

You are probably talking about contact smart cards:

These kind of cards are defined bt ISO/IEC 7816:

• ISO/IEC 7816-1 - Define physical characteristic
• ISO/IEC 7816-2 - Dimensions and location of the contacts
• ISO/IEC 7816-3 - Electrical interface and transmission protocols
• ISO/IEC 7816-4 - Organization, security and commands for interchange

Think of it as a stack, where the lowest numbers define physical/electrical aspects and the highest numbers define application aspects.

If a smartcard reader is ISO/IEC 7816 compliant, then it can communicate with any smartcard.

SIM cards, as far as the hardware is concerned, are just another form factor of common CC sized smart cards.