It's not unusual for disguises to seem paper-thin on stage, a dramatic convention to make allowance for limited props and budget. Willing Suspension of Disbelief is encouraged in this scenario.
In Ken Ludwig's Lend Me a Tenor, the theater manager's assistant Max disguises himself as the visiting tenor Tito Merelli after Tito is found dead in his hotel room. This works on the opera's audience, as they have never seen the real Tito and won't know the difference. However, Tito's Not Quite Dead, and is running around Cleveland in the same costume Max is wearing (Tito brought two identical costumes). Saunders, the manager, runs into Tito several times without realizing the costumed tenor is not Max, despite Max and Tito, even in costume, looking nothing alike.
In many stagings of As You Like It, Rosalind's "Ganymede" disguise is portrayed as this. In these versions, Orlando recognizes her at once, she realizes as much, and their subsequent "tutoring" scenes together become coy, humorous flirtations between the pair, who through role-playing can be frank with each other in ways a young couple of the time ordinarily couldn't.
In Holy Musical B@man!, Alfred is fired by Bruce, later returning as "O'Malley the Irish Butler" who is just Alfred wearing a hat and a ginger beard.