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Saved 57 bytes thanks to @Mr.Xcoder and, @pizzapants184 and @ovs
lambda a,b:max(f for f in a if f*(a.count(f)==b.count(f)<2)for f in a)
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Saved 5 bytes thanks to @Mr.Xcoder and @pizzapants184
lambda a,b:max(f for f in a if a.count(f)==b.count(f)<2)
Try it online!
Saved 7 bytes thanks to @Mr.Xcoder, @pizzapants184 and @ovs
lambda a,b:max(f*(a.count(f)==b.count(f)<2)for f in a)
lambda xa,yb:max(lf for lf in{*x}&{*y} a if xa.count(lf)+y==b.count(lf)<3<2)
lambda x,y:max(l for l in{*x}&{*y}if x.count(l)+y.count(l)<3)
