It fits, but I wouldn't equip a half-format camera (DX) with a full-format lens (FX).
Better to look for a suitable DX lens.

Of course this lens fits on a D5600, all Nikon DSLRs have the same mount. Only on the new Z cameras does the lens (without adapter) not fit.
@Stichus: Exactly the opposite should not be the case. An FX on DX makes sense, a DX on FX rather not.
It's better to buy FX lenses so that you don't have to buy all new lenses when you switch to FX later.
With a telephoto lens, the 1.5x crop is not a disadvantage but rather an advantage.
The 150-600mm "becomes" a 225-900mm, so you have even more zoom!
What you should look out for:
-> Wide-angle lenses for full format are usually not recommended for DX/APS-C, as they are usually not as wide-angled as they used to be, or they have a relatively high degree of distortion.
-> Older lenses do not have a built-in autofocus motor, but the newer Nikon DSLRs no longer have one in the body, so such lenses can only be used manually. But that does not apply here.

I have the Sigma 150-600mm Contemporary on my D7000, works perfectly. Unfortunately I can't say anything about the d5600.