There appear to be few similarities between the world of fairy that Oberon and Titania are monarchs of and the Elizabethan England in which Shakespeare wrote this play. In fact, any such comparison as this question is actually suggesting would have been very dangerous to draw, as Elizabeth I was a monarch who was named "the virgin Queen" and who made her singleness a deliberate political ploy in order to keep England as neutral as possible against the larger and more threatening powers in mainland Europe. Therefore drawing a comparison between the fairy land, where there is a King and a Queen, and Elizabethan England, which only had a Queen, would not have been a smart move on Shakespeare's part.
There is however a wider opposition that is created in the play between the world of the fairies in the forest and the world of humans set in Athens. The world of the fairies in the forest is characterised by magic, adventure and a lack of reason. Puck, for example, could be considered the chief mascot of this chaotic world, and he has the magical ability to transform himself into any shape he wants, as his taunts of the Mechanicals in Act III scene 1 demonstrate:
Sometime a horse I'll be, sometime a hound,
A hog, a headless bear, sometime a fire;
And neigh, and bark, and grunt, and roar, and burn,
Like horse, hound, hog, bear, fire, at every turn.
By contrast, the world of Athens is one that is defined in Act I scene 1 by the harshness and severity of the law, which could lead to the loss of Hermia's life if she does not agree to her father's demands to marry the man he would have her marry. If Athens is characterised by reason, the world of fairies is characterised by a kind of topsy-turvy chaos that impacts all who enter it. Interestingly, many productions have the characters of Hipolyta and Theseus play the characters of Titania and Oberon as well, drawing an interesting contrast between these two sets of lovers. Both at the beginning of the play have problems in their relationship, but both by the end are reconciled. If there is a comparison therefore between the fairy world and Elizabethan England it is that both are subject to the vagaries of love that show humans and fairies alike up to be fickle and changing in their affections.