You can control them both on one output. The issue may Be color order. Color order between the two strings are sometimes different. If they are then your colors would not match.
when you say “one output” do you mean having them on the same pin (join together) s on the ESP? or the string and the strip on the same ESP but both using different pins?
regardless, If I understand you properly, If I have an extra ESP I should use it to have better control ?