Adulteration of meat products is a serious problem in the modern society. Consumption of falsified meat products can be hazardous to health and/or lead to violating religious dietary principles. To identify such products, rapid and simple test systems for point-of-need detection are in demand along with complex laboratory methods. This study presents the first double lateral flow (immunochromatographic) test system, which allows simultaneous revealing two prevalent types of falsifications—undeclared addition of pork and chicken components to meat products. In the proposed test system, porcine myoglobin (MG) and chicken immunoglobulin Y (IgY) were used as specific biomarkers recognizable by antibodies. Within the optimization of the analysis, the concentrations of the immune reagents and regimes of their application on the working membrane were selected, which provided minimal limits of detection (LODs) for both analytes. The developed test system enables the detection of MG and IgY with the LODs of 10 and 12 ng/mL, respectively, which accords to addition of 0.1% of the undeclared meat compounds. The applicability of the test system to control the composition of raw meat mixtures and cooked food products was confirmed. The developed approach can be considered as a promising tool for monitoring composition of meat products.