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Barclays HP's show up the same, so generally the answer is yes(although I nor anyone else can guarantee this). I've only seen synchrony uniquely code HP's to each card generally from experience.
However, getting barclays to approve 2 cards in the same day will probably be the struggle
I tried that once. Second application was held 10 days, they then pulled a different credit bureau. FICO score on second bureau was 9 points higher, in spite of the first Barclay card already appearing on reports. Then they denied the second application.
But you might have completely different results. Perhaps...
They almost never approve someone for 2 accounts in one day anyway, so the question is kind of moot, but IF you can get two instant approvals than yes the HPs would probably be merged (no guarantee).
However the chances are pretty low that this will occur. They tend to want history on one account before issuing another. Barclay's was years ago much more lax and would give people 4 approvals in one day. But now, if you apply for a second card before 3-6 months with the first card, there's about a 98% chance of having to recon, and probably not getting approved. YMMV.