This is one of those slightly ambiguous things that people may start arguing about without reaching any definite conclusions, so I may be opening a Pandora’s box here. But for what it’s worth: no, the Euler-Lagrange equation is not a physical law. I have never seen it referred to as a physical law (explicitly or in an implied way) in any textbook or other authoritative source, and this is consistent with my own understanding of what a physical law is.
As a side remark, your own definition of a physical law is wrong. Physical laws are simply statements that describe some aspect of how the universe behaves and are found through experiment to be correct (within some possibly limited regime of parameters or circumstances). Examples include Newton’s laws of motion, Schrodinger’s equation, the ideal gas law, Coulomb’s law, Maxwell’s equations, etc. Physical laws need not necessarily mention time, position, or involve tensors. But they have to say something specific about the physical universe! For example, $2+2=4$ is not a physical law, and neither is the formula
(fg)’ = f’g+fg’
for the derivative of a product of two functions.
As for the Euler-Lagrange equation, what it is is a general mathematical device that enables us to encode a qualitative statement about a function $q(x,t)$ - the fact that it is a stationary element for a least action principle - in an explicit way as a differential relationship satisfied by the function. It is of course very important in physics, since it turns out that many physical laws can in fact be formulated as least action principles, but note that the Euler-Lagrange equation holds true for action functional associated with arbitrary Lagrangians, including those that have no physical meaning. So, at the end of the day the EL equation has no more claim to be called a physical law than the chain rule, the fundamental theorem of calculus, or other general mathematical results of this sort that get used all the time in math and physics.
Perhaps the most that one can say is that there are specific Lagrangians such that when one writes down the EL equation for that Lagrangian one gets a differential equation that is some well-known physical law - that I can certainly agree with. But to say that the EL equation itself is a physical law would be quite misleading, and, well, simply not true.