Label     Type
foo       B
bar       A
baz       _(empty cell)
qux       B

totalX    quantity
totalY    quantity

Type    X      Y
A       2      5
B       1      3
C       3      4

=sum(iferror(arrayformula(if(B2:B="";"";iferror(if(match(B2:B;'Sheet2'!A2:A4;0);vlookup(B2:B;'Sheet2'!A1:C4;{2}*sign(row(B2:B));false)))))))

=sum(iferror(arrayformula(if(B2:B="";"";iferror(if(match(B2:B;'Sheet2'!A2:A4;0);vlookup(B2:B;'Sheet2'!A1:C4;{3}*sign(row(B2:B));false)))))))