Why does Snug, who plays the lion's part, make a fuss about proclaiming his true identity in A Midsummer Night's Dream?

In Act 1 Scene 2 of A Midsummer Night’s Dream a group of tradesmen from Athens decide to put on a play for Duke Theseus to celebrate his wedding. The play they choose is Pyramus and Thisbe, a story of a couple who is in love but whose parents don’t want them to marry. They decide to run away together but when Thisbe gets to the meeting place, a lion frightens her away. When Pyramus arrives he thinks Thisbe has been killed by the lion and kills himself. Thisbe finds Pyramus’ body and kills herself.

As they plan the play, the actor Bottom wants to play every part. When he insists that he play the lion, he promises to play a terrifying lion. The director, Quince, tells him “An you should do it too terribly, you would fright the duchess and the ladies, that they would shriek; and that were enough to hang us all.” He is worried that if the lion is too scary it will frighten the women in the audience and the duke will punish them for that.

Clearly this threat of hanging hits home with Snug, the man playing the lion. In Act 5, Scene 1 when the play takes place, Snug takes great care to explain to the ladies in the audience that he is in fact not a real lion, that he is Snug, and that if he really were a lion this would be a bad place to try to cause trouble. This way, no one is frightened by the lion and the duke is not angered and will not punish them for the play.