The "more information" indeed comes from the assumption of a "common language" (i.e. a common & absolute scale in minds of all the voters). I should stress however that this point (which Arrow conveys) is being stressed not by the better-known proponents of Approval Voting (e.g. Brams & Fishburn), but rather by those of majority judgement (Balanski & Laraki). At a pure technical level, subsumming Approval Voting to majority judgement methods seems correct (see e.g. free PNAS article by Balanski and Laraki (2007) which summarizes their main ideas). The more subtle catch is that Approval Voting doesn't seem to depend on this notion of common language, while majority judgment methods with more than two grades for each candidates seem to need it to avoid some paradoxical interpretations. (As an aside, Arrow mentions by name Balanksi, but not Brams in that interview.)
In the absence of such an assumption, some cardinal methods can produce really counterintuitive results. This is not illustratable with Approval Voting (as far as I know), but with more grades it's easily shown; e.g. a book review by Edelman of Balinski (who is one of the main proponents of cardinal voting) & Laraki Majority Judgement (2011) gives the following example:
where the candidates are in Roman numerals (I-III) and the voters in Arabic (1-5), and there's American-school-like grading (A is best, F is worst). Under the assumption of common language, Balinski's method of picking II as the winner makes sense (he use the median, not the average of votes for the selection, and voter #3 is the median). On the other hand, if we don't assume a common [grading] language, then I is preferred to II by 4 of 5 voters, under the traditional social choice theory.
The interpretation(s) of approve/disapprove in Approval Voting have been at least 3 (in other words, that highlighted by Arrow from Balinski is not the only one), and these entail a complex discussion of strategic voting:
With regard to the meaning
of “approval” there appears to be a fundamental split among points-of-view. Some
presume that each voter actually has an underlying ranking (possibly weak) of the
alternatives, and in choosing an approved ballot must somehow compress several distinct
levels of approval into exactly two levels. Others view the dichotomous ballot
as a direct reflection of a dichotomous primitive: each voter either likes or dislikes
each alternative, and is indifferent among those within either group. A third view presumes
that a voter has a ranking together with a line dividing those alternatives she
likes from those she dislikes; we will refer to such a line as a true zero. An assignment
of cardinal utilities might underlie the first view, or the third (if utilities can be
negative). [footnote: The authors of Approval Voting take two views, referring sometimes to an underlying ranking and at other
times speaking of the approved alternatives as those “acceptable” to the voter. The third view appears in Brams
and Sanver (2009), which proposes voting rules that use a ballot consisting of a ranking and a dividing line
both, and in Sanver (2010); see also the related Bucklin voting, fallback voting, and majoritarian compromise
discussed in Hoag and Hallett (1926) and Brams (2008).]
A strategic analysis of approval voting cannot easily be disentangled from this
more philosophical matter of what it means to approve an alternative. If approval is a
primitive, then each voter has only one sincere ballot, but lacks any incentive to vote
insincerely. For a voter with an underlying ranking, it is clearly never strategically
advantageous to approve an alternative without also approving all others that you like
as well or better, so deciding on an approval ballot amounts to choosing “where to
draw the line.” If that line has no intrinsic meaning, there is no basis on which to
discriminate between a sincere ballot and an insincere one; one might argue that all
ballots are strategic, or that none are. If a voter has both a ranking and a line with
intrinsic meaning as a true zero, then any ballot drawing the line somewhere else might
be classified as insincere—and such a voter might have a strategic incentive to cast
such a ballot. [footnote: Marking a voter’s true zero is not the only way to ascribe intrinsic meaning to the location of “the line.” For a
voter who assigns a cardinal utility to each alternative, the mean utility value serves as a sort of relative zero
(quite possibly different from that voter’s “true” zero, if she has one) and arguments have been made (in Brams
and Fishburn, 2007) for drawing the line there. Duddy et al. (2013) show that drawing the line at the mean
maximizes a measure of total separation between the approved and unapproved groups. Laslier (2009) argues
that it is strategically advantageous to draw the line near the utility (to the individual voter) of the expected
winner, and that voters tend to behave this way.]
And another interesting observation:
Moreover, if we relax the SCF [social choice function] notion by allowing voters to express indifference among
alternatives, approval voting becomes an SCF—in fact it coincides both with Borda and
with all Condorcet extensions, when these rules are suitably adapted to handle ballots
with many indifferences.
Both quotes from Zwicker's Introduction to the Theory of Voting (2016).
And to be fair to Balanski & Laraki, in their book they have a section (18.3) that discusses how approval voting may be cast "approval judgement", in terms of formulating the questions on the ballot as to suggest a common grading scale etc.
When approval voting is practiced as a majority judgment, a language of two
words must be formulated that makes clear the evaluations are absolute grades.
To distinguish it from its traditional practice we call it approval judgment. [...]
So to nitpick, Arrow was basically talking of the latter.
