Based on my understanding, the Higgs field is a energy field which possesses the Higgs Boson which interact with matter to "give" them mass. However if all the electrons in the universe are having the same mass, does this mean that all the electrons in the universe interact with the same number of Higgs Boson particles? If so, how does it remain consistent?
The Higgs field is a quantum field. The Higgs boson discovered at the LHC is one of the excitations of this field but is not directly related to particle masses - the masses are due to an interaction with the Higgs field not the Higgs boson.
All elementary particles are described as excitations of their associated quantum field. For example there is an electron quantum field and electrons are described as excitations of this field. All electrons have identical properties because they are all excitations of the same field and the properties are determined by the field.
The mass of the electron is determined by an interaction between the Higgs field and the electron field (this interaction is called a Yukawa coupling) and this has a certain value that varies between different types of fields but has a fixed value for any particular field.
And this is why all electrons have the same mass. The mass is determined by the Yukawa coupling between the Higgs and electron quantum fields, and all electrons get the same value of this mass. Likewise if we consider muons, all muons get their mass from the Yukawa coupling between the Higgs and muon fields, which is different from the Higgs-electron field coupling. So all muons have the same mass but this mass is different from the electron mass.
The Higgs boson tends to be mentioned in popular science articles as if it was the cause of particles masses, but it's important to understand that this is not the case. The masses are the result of interactions with the Higgs field and the Higgs boson is just a side effect of the Higgs field's existence.