Carrying your own smart cards and smart card reader may work for some use cases but I'm sure you can see why a small key attached to your key chain is a better solution in most cases.
There are readers the size of a thumb drive, cards themselves are the standard credit card size or even SIM card size if not contactless. It's not the real issue here, there are a few more important ones such as lack of desktop browser support for U2F NFC use case. U2F applet works fine for me on Android though.