Is it correct to say that an integer is congruent with a p-adic number $\text{mod }p^k$? Is there a standard way to describe a class of $p$-adic numbers that share the same final digits? E.g., if $x$ is a $p$-adic number and the last four digits of its $p$-adic representation are $\ldots abcd$, can we say that $x \equiv {c*p + d} \pmod {p^2}$?
Furthermore, what if $x$ is a $p$-adic fraction, and its last four digits are $\ldots ab.cd$? Surely we can't say $x \equiv \frac{c*p+d}{p^2} \pmod {p^{-2}}$?

The analogy with ordinary integers and rational numbers works well.
To your first question: yes, you can totally say that. Just like with integers, $x \equiv y \pmod{m}$ means nothing more nor less than $m$ divides $x-y$, and “divides” means that $x-y=mk$ for some integer $k$.
In your example, $p^2$ divides the difference between the given number and $cp+d$, because that difference is a $p$-adic integer without a free term and a multiple of $p$ term. All the terms have $p^2$ and beyond.
To your second question, consider what you’d say with ordinary rational numbers. If $q=17.25$ we don’t say that $q$ is congruent to $0.25$ modulo $\frac{1}{100}$. We can say that $q$ is congruent to $0.25$ modulo $1$, which means “up to adding some integer”. The same is true in the field of $p$-adic numbers.
The rational numbers $\mathbb{Q}$ and the field $\mathbb{Q}_p$ of $p$-adic numbers are fields, so there’s no point in talking about congruence in the same way we do with rings such as the integers $\mathbb{Z}$ or the $p$-adic integers $\mathbb{Z}_p$. What we can say is that two elements differ by an integer.
