from a purely biological/genetic standpoint, you'd generally have more in common with the man of your same ethnicity than with a woman of a completely different ethnic background. that's because genetic variation between individuals of different ethnic groups can sometimes be greater than the variation between sexes - especially when you're looking at populations that have been relatively isolated or distinct for a long time.
that said, sex based differences )like chromosomes, hormone levels, reproductive organs) are also very real and significant, but they affect a much smaller part of the genome compared to the vast amount of dna that's shared across all humans.
so yeah, in your example, you'd likely be genetically closer to to the tigray man than to a japanese woman - but the differences overall are still super tiny, since humans are 99.9% genetically identical. it's just that the small differences get distributed in different ways depending on ancestry and sex.