In many standard texts I have seen the approach of taking any point connected with Earth in a circuit as 0 potential. Obviously, there is nothing special with 0 as long as we are concerned with potentials except that it is standard potential for infinity when there are only finite charges. I have seen a lot of question assigning same potential to both the Earth and infinity: such as this one 'A conducting sphere is initially charged with charge q. The sphere is then placed in a uniformly charged hollow sphere having charge q2 . The solid sphere in the interior is then connected to Earth with a conducting wire.There are no other charge distributions. Find the final charge on the sphere.'
We can take any point of our choice to be the zero potential. When we are working with circuits down on Earth we usually take the ground on Earth as the zero potential. When we are doing electrostatics or electrodynamics we usually take infinity to have the zero potential. It's just a matter of choice for the specific situation.
Why do we assume that the Earth and Infinity are at the same potential?
Short answer: we don't.
However, for different problems we do assign zero potential to infinite distance from objects of interest and to Earth (or anything else convenient).
For example, in computing the escape velocity of this earth this is typically done. In other sorts of problems involving Gravity, for example, Earth's surface is taken as 0 potential so that simple equations such as potential energy can be represented by: $m g h$.
With problems in electricity, we assign zero potential to any convenient location for the problem. In a circuit, such as used in a cell phone, the zero potential is nothing more than a common conductor considered as "ground" for the circuit. However, if I were to have another device, say it is wired to a grounding rod, and I call that zero potential, I could easily (sometimes) measure a potential difference between these two zero potentials (i.e. cell phone and grounding rod). This is because there are other electrical (e.g. static charges on phone) effects involved however both points are valid zero potentials for the problems they are solving.
Obviously, we all know that it is only the potential difference that is realistically involved in these problems. We can add to the absolute potential any finite value we want, whether positive or negative or zero, and the physics that we measure, observe, and work with does not change.
It is not clear whether you are asking about assigning zero potential to infinity and earth simultaneously or not. However, you can do that and even use those definitions in the same problem arena. Although you are assigning zero potential to infinity and Earth simultaneously you are not assigning it to the same type of potential.
A simple example with equipotential lines, which include infinity by construction:
The electric potential of a dipole show mirror symmetry about the center point of the dipole. They are everywhere perpendicular to the electric field lines.
The straight line is zero potential by the addition of the two charges, +1/r and -1/r, and is common all the way to infinity.
The problem at hand decides the zero of the potential, and there is no contradiction to having one at infinity as this simple example shows by the infinite zero potential points on the line.
This illustrates that it is the boundary conditions of the specific problem that decide where the zeros of the potential are, and there is no problem with having a zero at infinity and zeros at close by.points.
You can consider the earth to be at zero potential because it can be thought of as an infinite source and sink.
This means that if you apply positive volts, then the current will flow into the ground and if you apply negative volts, current will flow from the ground.
You should also know that the potential of a sphere drops at $1/r$ outside the sphere. Since this is the only charge in the universe therefore the potential at infinity is also zero.