I can’t speak to what data sharing the five eyes countries do, but i will note none of them care about dual citizenships. I’ve definitely entered UK, US, NZ, AU, SG with different citizenships (based on where I was flying out from). The US only noticed because I had forgotten I filed an ESTA in the other citizenship. Presented it and they were fine, despite me flying there using the other citizenship (this was a year or two ago, I don’t know what their policies are today).

Definitely SG doesn’t care what passport you use so long as at the immigration desk it’s the same one on arrival and departure. Which is kinda ironic given many Malaysians use it often for a dual passport change point.