Is there strong interaction between electrons?

The strong interaction that keeps protons together is a different kind of force (the strong nuclear force) which does not affect electrons. Electrons don't feel the strong force. They only feel the electromagnetic force and the left-handed ones also feel the weak nuclear force, which converts electrons into neutrinos. As a result, even if two electrons collide at high energy they will not cling together. There is no other force that can take over as in the case of protons.

Instead, the energy exchange may be large enough to produce extra particles. In a sense this is what was done at LEP (Large Electron-Positron collider), accept that the collisions occur between electrons and anti-electrons (positrons), which would attract each other and can annihilate each other.

The strong nuclear force is described by the theory of quantum chromodynamics (QCD). However, at low energies, such as where protons attracts each other, it is rather difficult to work with this theory because it is too nonlinear.

Strong interaction refers to a different sense of charge instead of electrostatic charge. At least that is to talk about the most direct use of that interaction. There are much, much weaker corrections that have that as an intermediate interaction (virtual quarks). The joining is Cooper Pairs in superconductors. Look that up to see how it is mediated.