Indeed, with the commencement of numerous large-scale sky surveys, a vast number of new variable stars have been discovered.
For exoplanets, they might use designations like TOI, such as TOI 6883A. For variables previously listed in the GCVS, they carry established names.
For the current large-scale surveys, coordinates are typically used as names—e.g., ZTF J173841.97+703715.3, GDS_J1842008-021733, CSS_J102714.3+205943, WISEA J205237.60+742907.8. Alternatively, internal alphanumeric codes are used, like ZTF18abtvqrl.
For Gaia DR3, we use Gaia IDs, such as Gaia DR3 2148402530483653120.
For the numerous variables discovered by TESS, TIC numbers are generally adopted as prefixes, like TIC 239133160—though this isn’t absolute.
For UG-type variables reported to the CBAT, it’s customary to call them TCP (e.g., TCP J19181542+3158142). For the TNS, corresponding identifiers are used (though TNS currently seems to discourage submitting variable star discovery reports).
For new variables discovered by amateurs, practices often mirror those of projects like ZTF—e.g., my discovery STSP J020810.62+794842.2. Sometimes designations with numbers are used, like Barlad V48 or CzeV926. Occasionally, bright variables may adopt IDs from historical surveys (e.g., HD, TYC, BD, URAT), such as HD 54896, TYC 7302-850-1, SAO 59882.
However, since each variable star likely exists in multiple catalogs, they typically carry several names. For instance, HD 54896 lists 12 “Other names.”
As Jim noted earlier, given the sheer number of variables (and considering future projects like Gaia DR4, WFST, and LSST will discover many more), standardization of naming is clearly unrealistic. For such a vast quantity, specific names seem less critical.
P.S. Relevant information is noted in the VSX FAQ and Guidelines.