But isn’t the ‘other sheep’ in ‘another fold’ refer to the Jews vs Gentile Sheep of His?

No!

In fact, here a set of Verses proving irrefutably that both the Jew and Gentile Sheep of His (on earth, from the Church) are mentioned together as ONE FLOCK (implying that such a ‘popular claim is FALSE’):

Yes, BOTH the ‘believing Sheep of Israel’ and the ‘believing Sheep of the Gentiles’ belong to the SAME FOLD, namely sheepfold1, Proof Verses:

“solemnly testifying to BOTH JEWS and GREEKS of repentance toward God and FAITH in our Lord Jesus Christ… Be on guard for yourselves and for ALL the FLOCK, among which the Holy Spirit has made you OVERSEERS, to SHEPHERD the CHURCH of GOD which He purchased with His own blood. “I know that after my departure savage wolves will come in among you, not sparing the FLOCK;” – apostle Paul (Acts 20:21, 28 – 29, NASB)

So, the Sheepfold1 = His Church = His Bride = His Elect ONLY (1 Thessalonians 4:14 – 17).

Please note carefully these phrases in Verses above to realise this:

BOTH JEWS and GREEKS —-> FAITH —> ALL the FLOCK —-> OVERSEERS, to SHEPHERD the CHURCH of GOD —-> FLOCK  —-> which clearly proves that His CHURCH does NOT Consist of two different FOLDs (e.g. Jew vs Gentile as commonly erred, NO Verse says this) but only as ONE FOLD.

The Lost Sheep of Israel (Matthew 15:24) vs the Lost Sheep of the Gentiles (by Contrast in Matthew 10:5 – 6) only Speaks of the SAME FLOCK or ONE FOLD of His CHURCH when SAVED (as per Acts 20:21, 28 – 29) where God only sent them to the Gentile Lost Sheep after the Day of Pentecost due to Empowerment from Above (Acts 1:4 – 8).

In other words, the distinction between the Lost Sheep of Israel (Matthew 15:24) vs the Lost Sheep of the Gentiles (by Contrast in Matthew 10:5 – 6) who are going to ‘believe in Him on earth itself’ (“FAITH”, Acts 20:21 as a requirement in the definition of this FOLD) are each mentioned distinctly in Holy Scripture with respect to the Gospel being Preached to them before the Empowerment of His Holy Spirit (Matthew 15:24) vs after His Empowerment by His Holy Spirit on the Day of Pentecost (Acts 1:4 – 8).