Python:

• the tests framework is not properly used (missing describe/it)
• the input is confusingly named arr, even though it is a string

no random tests.
(check my solution) :P

verifies that provided argument is either an integer or a floating-point number

How this is integer -3.2 : expacts True

Could you explain how "12, 88, 816 and even 1180344 (and of course, 2360688 itself)" are also solutions of digPow(46288, 3)? I think you're not understanding what to do, the solution is unique or doesn't exist (in that case you have to return -1).

Without a stated upper limit on n and k

n doesn't need a limit, a single value is passed to you (as it is p), you have to work with that, nothing more. k is what you have to find.

Print the input.

-> "I don't think there's a situation when 6.e+2 would count as a number..."

In python (at least) : 6.e+2 is the float 600.0, so this notation has to be considered as a number, actually.

Error messages were not that clear when submitting the solution. Argument name why "arr"??? Function name could be something more obvious, not that it was hard to understand but as a good standard.

Is there a way to show what numbers fail in the regex pattern?
I tested my regex and 74 expression passed but 8 expression failed, I would like to know what numbers didn't pass the regex. I tested the regex and seems to look ok.

What about leading and trailing zero which are ignored by try/float()/except solutions?

There should be a test with only one dot since some regexp solutions consider it a valid float.

I think, it should be mentioned somewhere that there should be no spaces in notation. Otherwise '- 12' should be considered as integer.