Since when did European regulations require the ability to suppress alerts?
Since June 2011.
EASA first proposed a change to CS 25.1322 in NPA 2009-12 (Notice of Proposed Amendment) in November 2009:

CS-25 contains a certification specification (CS 25.1322) that dictates the colour of
warning, caution, advisory, and other message lights that are installed as annunciation
displays in the flight deck. As presently written, CS 25.1322 addresses visual alerts only in
the form of coloured lights installed in the flight deck. No specifications are stipulated to
cover new technologies or the use of alternate media, such as aural tones/voice that can
be more effective. CS 25.1322 is therefore considered outdated and does not address the
safety concerns associated with today’s display systems.

EASA then included these changes to CS-25 (the Certification Specifications for Large Aeroplanes) in Amendment 11 on 27 June 2011. The version of CS 25.1322 before (last amendment of 16 December 2010) shows this text:

CS 25.1322 Warning, caution, and
advisory lights
If warning, caution, or advisory lights are installed in
the cockpit, they must, unless otherwise approved by
the Agency, be –
(a) Red, for warning lights (lights indicating a
hazard, which may require immediate corrective
action);
(b) Amber, for caution lights (lights indicating
the possible need for future corrective action);
(c) Green, for safe operation lights; and
(d) Any other colour, including white, for lights
not described in sub-paragraphs (a) to (c) of this
paragraph, provided the colour differs sufficiently
from the colours prescribed in sub-paragraphs (a) to
(c) of this paragraph to avoid possible confusion.

(EASA CS-25 Amendment 10)
Afterwards, the text is very similar to the one in 14 CFR:

CS 25.1322 Flight Crew Alerting
(a) Flight crew alerts must:

provide the flight crew with the
information needed to:
(i) identify non-normal operation or
aeroplane system conditions, and
(ii) determine the appropriate actions,
if any;
be readily and easily detectable and
intelligible by the flight crew under all
foreseeable operating conditions, including
conditions where multiple alerts are provided;
(3) be removed when the alerting
condition no longer exists.

(b) Alerts must conform to the following
prioritisation hierarchy based on the urgency of flight
crew awareness and response:

Warning: For conditions that require
immediate flight crew awareness and immediate
flight crew response.
Caution: For conditions that require
immediate flight crew awareness and subsequent
flight crew response.
Advisory: For conditions that require
flight crew awareness and may require subsequent
flight crew response.

(c) Warning and Caution alerts must:

be prioritised within each category,
when necessary;
provide timely attention-getting cues
through at least two different senses by a
combination of aural, visual, or tactile indications;
permit each occurrence of the
attention-getting cues required by sub-paragraph
(c)(2) to be acknowledged and suppressed, unless
they are required to be continuous.

(d) The alert function must be designed to
minimise the effects of false and nuisance alerts. In
particular, it must be designed to:

prevent the presentation of an alert
when it is inappropriate or unnecessary;
provide a means to suppress an
attention-getting component of an alert caused by
a failure of the alerting function that interferes
with the flight crew’s ability to safely operate the
aeroplane. This means must not be readily
available to the flight crew so that it could be
operated inadvertently or by habitual reflexive
action. When an alert is suppressed, there must be
a clear and unmistakable annunciation to the flight
crew that the alert has been suppressed.

(e) Visual alert indications must:

conform to the following colour
convention:
(i) Red for Warning alert
indications.
(ii) Amber or yellow for Caution
alert indications.
(iii) Any colour except red or green
for Advisory alert indications.
use visual coding techniques, together
with other alerting function elements on the flight
deck, to distinguish between Warning, Caution
and Advisory alert indications, if they are
presented on monochromatic displays that are
incapable of conforming to the colour convention
in paragraph (e)(1).

(f) Use of the colours red, amber and yellow on
the flight deck for functions other than flight crew
alerting must be limited and must not adversely
affect flight crew alerting.

(EASA CS-25 Amendment 11)