No it is not necessary that the colliding particles 'stick' together after a totally inelastic collision, in the sense that a force must be applied to separate them again. Some kinetic energy is lost, which could be used to 'fuse' the two particles or form a bond between them. But this energy could otherwise be transformed into heat or permanently deform the particles without making them 'stick' together.
The essential requirement is that the relative velocity of separation becomes zero - ie the particles move together with the same velocity after the collision. Alternatively, "totally inelastic" means that in the centre of momentum frame of reference (in which the centre of mass of the particles is at rest) all of the kinetic energy is destroyed by the collision, transformed to internal energy.
This applies in all 3 dimensions. You cannot have a collision which is totally inelastic in one direction and only partially inelastic in another direction, because then the relative velocity after the collision would not be zero, which contradicts the definition of a "totally inelastic" collision.
In your problem, if after colliding the particles have the same velocity in the x-direction but different velocities in the y-direction then the relative velocity after the collision is not zero, so by definition the collision was not totally inelastic.