I would think one of the examples that most similarly mirrors the SC2 example would be tennis: while both players play on the same court with the same ball, each player chooses their racquet ahead of time and there are a variety of factors towards choosing the racquet and string that could provide benefits.
Here are some examples of individual factors chosen by the player for their racquet before coming to the match:
- the size of the racquet: larger racquets have larger sweet spots and provide more surface area to hit the ball, but they are heavier and harder to swing
- the type of string: tennis racquet string comes in a variety of natural and synthetic materials that offer different properties. One is the texture/shape of the string; "rough" strings with a polygonal shape provide more friction and thus more spin to the ball. Softer, fibrous strings give an increased "feel" and control on impact. Oftentimes players will choose a different type of string for the "mains" (the strings that run vertically on the racquet) and the "crosses" (the strings that run horizontally) for a blend of the two properties and to suit their playstyle.
- the tension of the string: when a racquet is strung, the strings can be placed under specific amounts of tension to alter the properties of the racquet. Stringing a racquet with higher tension gives the racquet more control as the string will deform and shift less on impact, while lower tension increases the trampoline like nature of the racquet and delivers more power.
Altogether, a player that wants to maximize their power might choose a larger, more loosely strung racquet than a player that wants to maximize their control and maneuverability.
These are some considerations for tennis (and any other strung racquet sport like badminton, etc) to be argued as an asymmetric game, although the factors are relatively small in comparison to something like different races in SC2. If the game was completely symmetrical, I'd expect both players to have to use a standardized racquet or have to exchange racquets at some point during the match.
Maybe a more drastic example would be motorsport divisions that are not "spec" (ie they all use the exact same chassis, engine, suspension, etc). Each racer is all racing the same course, but if they are in different cars will different capabilities, that would seem pretty asymmetric to me.