As an example i pull my last proof Proofing divisibility by $7$ . This is basicly an adaptation of similar proofs i found on youtube.
Nearly all of them use n=k in the assumption and n=k+1 in the proof.
So is it superfluous or is there some sense behind it? (like more complex proofs which are made easier by n=k)
As an example, where i think it would make sense, would be a proof about a converging series, where you rearrange terms in a finite series (with k), to be able to rearrange to then go to a infinite series.
In my opinion (which of course if fallible) I'd say that $\textit{once}$ you get used to induction, then you can proceed as you wish, either using this convention or ignoring it - and the better you get, the more likely you are to use your intuition as to whether to include it or not. Personally, I can't remember the last time I wrote the whole "Assume n=k holds, how about n=k+1) - although this is what I was taught at college (UK, high school USA).
In my first year at university, I was taught to think of induction like this: We haves a sentence about n, $P(n)$, that is either true or false. What we try to achieve is the idea that if $P(n)$ evaluates to true, then $P(n+1)$ evaluates to true as well. For example, we could have a statement: For all $n>3, n!>2^n$ and our proof by induction goes like this: If $n!>2^n$ then multiplying the lhs by $n+1$ and the rhs by $2$ we keep the inequality (why?) to find $(n+1)!>2^{n+1}$. This means that, if we know that $P(n)$ is true, then $P(n+1)$ is true as well. Since $4!=24>16=2^4$ we have $P(4) \implies P(5) \implies...$ for all $n>3$.
As you see, I don't bring in $k$. But that isn't to say that, when learning about induction, it isn't useful. At schools it is taught (I believe) to aid in understanding the different objects we call $n$ - at points in the proof we are concerned about a specific value of $n$ (for example the base case or special cases) other times, we have a range of $n$ (for example $n>3$). It allows you to do a lot of manipulation with this object $k$ without getting confused with your statement about $n$.
I personally feel like it is a matter of taste, does it make it easier for you to prove something? Then great. Does it make it easier for someone to understand your proof? Then great. Is there a hardcore, reason why it should be done? I don't know of one.