Did the Russian Navy have two different types of sailor suit, or was it a fashion thing, or is there some other reason?
No, the Russian Navy didn’t have two different types of sailor suit. All sailors wore a dark blue middy blouse in winter and a white one in summer, but both had the same pattern (without a tie like the British). Uniforms for the Guards’ Equipage (which included crews of the imperial yachts) were slightly different than the « regular » Navy, but they basically looked alike.
Technically, no one was allowed to wear a Navy (or Army) uniform without being formally enrolled in a Navy crew or a regiment. But there seems to have been some exceptions for young members of the Imperial Family… as I have seen photos of young Michael Alexandrovich, Boris & André Vladimirovich wearing a uniform from the Naval Guards Equipage (which provided crews for the Imperial yachts).
Some young grand dukes were enrolled in the Guards Equipage at a very young age though, and were legally entitled to wear its uniform. For instance, Emperor Nicholas II was enrolled in the Guards Equipage at birth. George never was but he did actual service in the Navy, so I guess the picture you are refering to is from that time. Tsarevich Alexis was enrolled in the Guard Equipage on the day he was baptised and did wear their uniform from a very young age. I don’t recall having seen him in a « British » style uniform.
As for Sandro & Xenia’s sons, none of them were enrolled in the Navy (or Army) as young children (as they weren’t grand dukes but only princes of the Imperial Blood). But the older boys – if not all of them – were later enrolled in the Naval Cadet School, whose uniform looks pretty much like the Navy one, and some Mikhailovichi can indeed be seen on photos wearing it. Then again…the exception seems to apply, as I have seen a photo showing Feodor & André in their official Naval Cadet School uniform, Dimitri, Rostislav & Vassili in a « British-style » uniform...but young Nikita in a Russian Navy uniform (with a cap from the « Tamara » on which his father once served). So…go figure!