Is 1 ppb equal to 1 μg/kg?
You are correct suggesting that 1 μg/kg implies 1 ppb, however the reverse is not true. For instance, 1 ppb can also be 1 nmol/mol, and the reader will never have a chance to deduce which one is it unless you explicitly define the usage of the "parts per something" in the text.
This clutters the manuscript with redundant notes and causes overall confusion.

IUPAC also lists all similar symbols (ppm, ppt, ppb etc.) as deprecated; from IUPAC's “Green Book” [1, p. 98]:


  Although ppm, ppb, ppt and alike are widely used in various applications of 
  analytical and environmental chemistry, it is suggested to abandon completely their use because of the ambiguities involved. These units are unnecessary and can be easily replaced by SI-compatible quantities such as pmol/mol (picomole per mole), which are unambiguous. The last column contains suggested replacements (similar replacements can be formulated as mg/g, μg/g, pg/g etc.).
  
  $$
\begin{array}{lllll}
\hline
\text{Name} &amp; \text{Symbol} &amp; \text{Value} &amp; \text{Examples} &amp; \text{Replacement} \\
\hline
\ldots &amp; &amp; &amp; &amp; \\
\text{part per billion} &amp; \text{ppb} &amp; 10^{-9} &amp; \text{The air quality standard for ozone is a} &amp; \pu{nmol/mol} \\
  &amp; &amp; &amp; \text{volume fraction of}~\varphi = 120~\text{ppb} &amp; \\ 
\ldots &amp; &amp; &amp; &amp; \\
\hline
\end{array}
$$


References


IUPAC “Green Book” Quantities, Units, and Symbols in Physical Chemistry, 3rd ed.; Cohen, R. E., Mills, I., Eds.; IUPAC Recommendations; RSC Pub: Cambridge, UK, 2007. (PDF)