Actually the standard hydrogen electrode is not always the reference. In much work a more convenient reference is used.
One of the most common is the silver-silver chloride electrode (or just silver chloride electrode, https://en.wikipedia.org/wiki/Silver_chloride_electrode), in which metallic silver and sparingly soluble silver chloride are equilibrated with a standard solution of a soluble chloride salt. As we can see from the illustration in the referred article, such electrodes based on sparingly soluble salts are in fact more convenient for common laboratory use than the standard hydrogen electrode.
Of course, the silver-silver chloride electrode has a different equilibrium potential versus the standard hydrogen electrode (SHE), specifically:
Thus if we are witking with a silver-silver chloride reference, equilibrium potential numbers are shifted by 0.230 V. Copper reduction occurs at +0.34 V versus a standard hydrogen electrode, but +0.11 V versus a silver-silver chloride electrode. But since all the numbers are shifted by the same amount, the difference between reduction potentials, which is what's physically important for overall reactions, stays the same.
For more on reference electrodes see https://en.wikipedia.org/wiki/Reference_electrode.