Two separate saves (if the strongest/most recent spell ends).
While both spells are maintained, the most potent or most recent one is active.
The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect - such as the highest bonus - from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.
Ensnaring strike (PHB, p. 237) reads
While restrained by this spell, the target takes 1d6 piercing damage at the start of each of its turns. The creature restrained can make a Strength check against your spell save DC. On a success, the target is freed.
So, while both spells are maintained, the strongest ensnaring strike is active, restraining and damaging the target, while the weakest one is suppressed (but still on-going, and its caster is still maintaining concentration). When the target succeeds on a save, the target is freed from that spell. The spell hasn't finished (lasts 1 minute), but its caster can dismiss it at any time, since it is a concentration spell, and is now useless (no longer doing damage or restraining a target, it's just wasting concentration).
If the weaker spell is still being maintained (less than a minute has passed since it was cast, and the caster has not lost concentration) when the most potent spell ends (its caster dropped concentration, or its duration ran out), the weaker spell is now active, restraining and damaging the target, which requires another save.
Someone has pointed out (in a now deleted comment below) how the spell says the target is freed and not the spell ends. You can argue that "getting freed" would release you from any other restraint, but I think it makes no sense that the target gets freed from all restraints and incapacitates affecting it. If you had your Brawler Barbarian grappling and restraining a target, would his getting rid of Ensnaring Strike also free him from the Barbarian? It seems clear to me the idea is that making your save frees you from the effects of this spell, and nothing else.