This is a painful way to do it, and it's not a good way, but for the number you have above, you could do

"951.534868sec" | into duration | to text | parse '{min}min {sec}sec {ms}ms' | $"00:($in.min.0):($in.sec.0).($in.ms.0)"
00:15:51.534

We already know those components. One would think we'd have a parameter to spit that out but I don't think we do.

Why its possible to apply to text on a duration type, but not into string?

G:\〉"951.534868sec" | into duration | to text                                                      03/06/2023 11:39:59 15min 51sec 534ms
G:\〉"951.534868sec" | into duration | into string                                                  03/06/2023 11:40:06 Error: nu::shell::cant_convert (link)

  × Can't convert to string.
   ╭─[entry #6:1:1]
 1 │ "951.534868sec" | into duration | into string
   ·                                   ─────┬─────
   ·                                        ╰── can't convert duration to string
   ╰────

My understanting is that every type should be convertible to string.

It's funny that duration has a lot of useful operations, but it can't be used in many scenarios that I can think of. After manipulating a duration one needs to output it, and the default format it not appropriate in most situations.

Have you considered adding more operations to it, like formating and extracting individual fields?

Converting an object to a string just to parse it again, does not looks like the "nu way"! (I would expect to do this in bash...)