IUPAC name of H[AuCl4]
The authoritative source Nomenclature of Inorganic Chemistry, IUPAC Recommendations 2005 (Red Book) lists $\ce{H[AuCl4]}$ as an example of a salt in the subsection IR-4.4.3.4 Generalized salt formulae [1, pp. 61–62].
Further, introduction to the section IR-8 Inorganic Acids and Derivatives underlines that IUPAC nomenclature is established from composition and structure, not properties, hence the word &quot;acid&quot; should not be used in systematic names [1, p. 124]:

Certain inorganic and simple carbon-containing compounds are commonly given non-
systematic or semi-systematic names containing the word ‘acid’. Examples are boric acid or
orthoboric acid, metaboric acid, phosphoric acid, diphosphoric acid, cyclo-triphosphoric
acid, catena-triphosphoric acid, dithionous acid, peroxodisulfuric acid or peroxydisulfuric acid, etc. These names are unique in modern nomenclature in that, interpreted literally, they describe a particular chemical property of the compounds in question. Systematic names are otherwise based solely on composition and structure.
All such acids may also be given structure-based systematic names using principles
already described in preceding chapters on substitutive and additive nomenclature, so in that respect the ‘acid’-containing names are superfluous. Furthermore, many species which
would be classified as acids based on their chemical properties are never named as such,
e.g. aqua ions such as hexaaquaaluminium(3+), and hydrides and derivatives such as
ammonium, hydrogen sulfide (sulfane), etc. The term ‘acid’ is thus not used consistently.
Based on these considerations, the use of the word ‘acid’ in any new name in inorganic
nomenclature is discouraged. However, a number of the existing ‘acid’ names are so
commonly used (sulfuric acid, perchloric acid, etc.) that it would be unrealistic to suggest replacing them altogether by systematic alternatives.

According to Wikipedia, $\ce{H[AuCl4]}$ exists in aqueous solutions as square planar $\ce{AuCl4-}$ anion, which is also retained upon crystallization as tetrahydrate, but there is a rapid
exchange of all protons in the unit cell [2]. For compounds of this class where the structural aspects of protons are either not essential or cannot be precisely established, a hydrogen name can be used. From the section IR-8.4 HYDROGEN NAMES [1, pp. 134–136]:

An alternative nomenclature for hydrogen-containing compounds and ions is described here. The word ‘hydrogen’, with a multiplicative preﬁx if relevant, is joined (with no space) to an anion name formed by additive nomenclature and placed within appropriate enclosing marks (see Section IR-2.2). This construction is followed (again with no space) by a charge number indicating the total charge of the species or structural unit being named (except for neutral species/units).
Hydrogen names are useful when the connectivity (the positions of attachment of the
hydrons) in a hydron-containing compound or ion is unknown or not speciﬁed (i.e. when
which of two or more tautomers is not specified, or when one does not wish to specify
a complex connectivity, such as in network compounds).
[…]
Examples:
[…]

$\ce{H4[Fe(CN)6]}$
tetrahydrogen(hexacyanidoferrate)
$\ce{H2[PtCl6]·2H2O}$
dihydrogen(hexachloridoplatinate)—water (1/2)

[…]
The strict deﬁnition of hydrogen names proposed here is meant to eliminate such
confusion by imposing the requirements:
(i) that ‘hydrogen’ be attached to the rest of the name,
(ii) that the number of hydrogens must be speciﬁed by a multiplicative preﬁx,
(iii) that the anionic part be placed in enclosing marks, and
(iv) that the charge of the total structure being named is speciﬁed.
Hydrogen names constructed in this way cannot be mistaken for other types of name.

To sum it all up, I think the IUPAC name for $\ce{H[AuCl4]}$ is hydrogen(tetrachloridoaurate).
Note that new recommendations also suggest using chlorido instead of chloro for anionic ligands. Also, in names formal oxidation states are enclosed in parentheses, not square brackets.
Wikipedia might not always mention IUPAC names, and even those which are listed as such should not be automatically trusted. Also, acceptance of systematic names in chemical community is inertial; not every systematic name is immediately replacing the traditional one which everyone is familiar with. I suspect this is the case as to why chloroauric acid still gets a pass.
References

IUPAC. Nomenclature of Inorganic Chemistry, IUPAC Recommendations 2005 (the “Red Book”), 1st ed.; Connelly, N. G., Damhus, T., Hartshorn, R. M., Hutton, A. T., Eds.; RSC Publishing: Cambridge, UK, 2005. ISBN 978-0-85404-438-2.
O’Reilly, D. E.; Peterson, E. M.; Scheie, C. E.; Williams, J. M. Nuclear Magnetic Resonance of the Aquated Proton. II. Chloroauric Acid Tetrahydrate. Phase Transitions and Molecular Motion. J. Chem. Phys. 1971, 55 (12), 5629–5635. DOI: 10.1063/1.1675731.