Plants cannot distinguish between different brands or marketing formulations of fertilizers. Fertilizer isn't plant food, but simply a mixture of nutrients that plant roots absorb to sustain plant metabolism and growth. Plants make their own food in their green leaves.
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A fertilizer with a 10-10-10 formulation is typically called a balanced fertilizer because the three main nutrient components—nitrogen, phosphorus and potassium, respectively—are in equal weight percentages. Balanced fertilizers are used to supply nutrients to a wide array of edible and ornamental woody and herbaceous plants.
Lawns given 10-10-10 fertilizer will still benefit from the addition of nutrients to the soil. Turf grass is grown for its lush green leaf blades, so nitrogen nutrients are often more important to creating a dense, fast-growing lawn. The grass will also utilize phosphorus and potassium, but perhaps not as heavily as the nitrogen available.
Depending on the topsoil's pH, organic matter components and parent material (sand or clay, for example), application of a balanced 10-10-10 fertilizer may not supply ample nitrogen for long-term dense growth of the grass. It may need additional applications to provide enough nitrogen based on your local soil's characteristics.