Do the natural numbers whose binary (representations) strings share the same particular binary substring also share any mathematical property? One obvious example: when a single one-bit substring in the first binary place of any natural number is "1" that number is "odd" and when it is "0" that number is even. But my question refers generally to repeated multi-bit substrings among different distances of bits. For example consider these following numbers:
\begin{equation*} 13 \rightarrow \begin{array} {|r|r|r|r|}\hline \color{red}1&\color{red}1&\color{red}0&\color{red}1 \\ \hline \end{array} \end{equation*} \begin{equation*} 77 \rightarrow \begin{array} {|r|r|r|r|r|r|r|}\hline 1&0&0&\color{red}1&\color{red}1&\color{red}0&\color{red}1\\ \hline \end{array} \end{equation*} \begin{equation*} 893 \rightarrow \begin{array} {|r|r|r|r|r|r|r|r|r|r|}\hline \color{red}1&\color{red}1&\color{red}0&\color{red}1&1&1&\color{red}1&\color{red}1&\color{red}0&\color{red}1\\ \hline \end{array} \\ \end{equation*} \begin{equation*} 108237 \rightarrow \begin{array} {|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\hline \color{red}1&\color{red}1&\color{red}0&\color{red}1&0&0&\color{red}1&\color{red}1&\color{red}0&\color{red}1&1&0&0&\color{red}1&\color{red}1&\color{red}0&\color{red}1 \\ \hline \end{array} \\ \end{equation*}
Is there any mathematically meaningful connection among the numbers of "13", "77", "893", "108237"?
It somewhat depends on what you mean by "mathematical meaningful connection among the numbers..."
However, I would say in general the answer to your question is no. Consider that for any given binary string, as $n \to \infty$, the proportion of integers containing that binary string approaches $1$. So in some sense almost all integers contain the given binary string as a substring.
As you note, if you know about specific bits in specific positions, that gives you more interesting information.