Why doesn't the binary fraction representation match the decimal fraction representation?
No, that's not how binary fractions work.
A decimal such as $0.13$ represents
$$0.13_{\mathrm{dec}} = 1\times 10^{-1} + 3\times10^{-2} = \frac{13}{100}\,.$$

Similarly, a binary fraction such as $0.1101$ represents
$$0.1101_{\mathrm{bin}} = 1\times 2^{-1} + 1\times 2^{-2} + 0\times 2^{-3} + 1\times2^{-4} = \frac{13}{16}\,.$$
$1101_{\mathrm{bin}} = 13_{\mathrm{dec}}$ but
$$0.1101_{\mathrm{bin}} = \frac{13}{16}\neq\frac{13}{100}=0.13_{\mathrm{dec}}\,.$$

Another way to see that you can't convert fractions into binary by forgetting that they're fractions is to note that $0.3=0.30$, but $3\neq 30$. So it can't possibly be that you convert $0.3$ in decimal to binary by saying "$3$ in binary is $11$, so $0.3$ in binary is $0.11$", since $0.3=0.30$ in decimal, so they should be equal in binary, too. But $30$ in binary is $11110$.