It can matter. Batteries are called out by specific manufacturer and model number in safety listing reports. There are single-fault tests that are conducted in the charging circuit and in the load circuit. The battery must not become a fire/explosion hazard when subjected to these faults. A replacement battery may not have the same performance as the device in the listing report. Here is a good example of what might typically be driving this sort of product labeling - the following is from information technology standard IEC/ANSI/UL 60950:

4.3.8 Batteries
NOTE 1 Requirements for markings or instructions are given in 1.7.15.
Equipment containing batteries shall be designed to reduce the risk of fire, explosion and
chemical leaks under normal conditions and after a single fault in the equipment (see 1.4.14),
including a fault in circuitry within the equipment battery pack. For USER-replaceable batteries, the
design shall reduce the likelihood of reverse polarity installation if this would create a hazard.
Battery circuits shall be designed so that:
– the output characteristics of a battery charging circuit are compatible with its
rechargeable battery; and
– for non-rechargeable batteries, discharging at a rate exceeding the battery
manufacturer’s recommendations, and unintentional charging, are prevented; and
– for rechargeable batteries, charging and discharging at a rate exceeding the battery
manufacturer’s recommendations, and reversed charging, are prevented.
NOTE 2 Reversed charging of a rechargeable battery occurs when the polarity of the charging circuit is reversed, aiding the discharge
of the battery.
Compliance is checked by inspection and by evaluation of the data provided by the equipment
manufacturer and battery manufacturer for charging and discharging rates.
When appropriate data is not available, compliance is checked by test. However, batteries that
are inherently safe for the conditions given are not tested under those conditions. Consumer
grade, non-rechargeable carbon-zinc or alkaline batteries are considered safe under
short-circuiting conditions and therefore are not tested for discharge; nor are such batteries
tested for leakage under storage conditions.
A new non-rechargeable battery or fully charged rechargeable battery provided with, or
recommended by the manufacturer for use with, the equipment shall be used for each of the
following tests:
– for evaluating the overcharging of a rechargeable battery, a battery is charged for a
period of 7 h under each of the following conditions in turn:
• with the battery charging circuit adjusted for its maximum charging rate (if
such an adjustment exists); followed by
• any single component failure that is likely to occur in the charging circuit and
which would result in overcharging of the battery; and
– for evaluating the unintentional charging of a non-rechargeable battery, a battery is
charged for 7 h with any single component failure that is likely to occur and which
would result in unintentional charging of the battery; and
APRIL 1, 2003 CAN/CSA-C22.2 NO. 60950-1-03 ¨ UL 60950-1 156
P.1
– for evaluating the reversed charging of a rechargeable battery, a battery is charged
for 7 h with any single component failure that is likely to occur and which would result
in reversed charging of the battery; and
– for evaluating an excessive discharging rate for any battery, a battery is subjected
to rapid discharge by open-circuiting or short-circuiting any current-limiting or voltagelimiting
components in the load circuit of the battery under test.
NOTE 3 Some of the tests specified can be hazardous to the persons carrying them out; all appropriate measures to protect personnel
against possible chemical or explosive hazards should be taken.
These tests shall not result in any of the following:
– chemical leaks caused by cracking, rupturing or bursting of the battery jacket, if
such leakage could adversely affect required insulation; or
– explosion of the battery, if such explosion could result in injury to a USER; or
– emission of flame or expulsion of molten metal to the outside of the equipment
ENCLOSURE.
After completion of the tests, the equipment is subjected to the electric strength tests of 5.3.8.2.