Same issue in en-GB locale, dates are displayed in US format.
Still an issue. It began about when the chat redesign was done. Locale is en_GB.
It's affecting both dates and times – as well as the broken muddle-endian date format, ) see abominations such as ”02:00 PM” – that should be “2:00pm” or, preferably, “14:00”. (I'm used to that leading 0 indicating 24h clock.)
(I would prefer ISO 8601 to be used everywhere. That way, there'd be no ambiguity.)

New year, same problem! Been annoying me for over a year so I dare say it's time I complain! :P
On the flipside, I'm getting a 24-hour clock in the overlay while in the US locale. All other time formats in Steam seem to be correctly 12-hour except that one (the in-game overlay clock).
This wrong behavior is still present as of today.
mm/dd/yyyy is superior and valve is hoping you realize this.
This issue still persists. There is no way to change time/date format in Steam Linux client. It is stuck with 12-hour middle-endian format.
LANG=fi_FI.UTF-8
LANGUAGE=fi
LC_CTYPE="fi_FI.UTF-8"
LC_NUMERIC="fi_FI.UTF-8"
LC_TIME="fi_FI.UTF-8"
LC_COLLATE="fi_FI.UTF-8"
LC_MONETARY="fi_FI.UTF-8"
LC_MESSAGES="fi_FI.UTF-8"
LC_PAPER="fi_FI.UTF-8"
LC_NAME="fi_FI.UTF-8"
LC_ADDRESS="fi_FI.UTF-8"
LC_TELEPHONE="fi_FI.UTF-8"
LC_MEASUREMENT="fi_FI.UTF-8"
LC_IDENTIFICATION="fi_FI.UTF-8"
LC_ALL=fi_FI.UTF-8
This appears to finally be fixed as of the March 28 build:

Unfortunately I'm seeing this once again. mm/dd/yy in chat timestamps and MM dd, yyyy elsewhere. My locale, including LC_TIME, is en_GB.UTF-8; these formats should be dd/mm/yy and dd MM yyyy. Times are reasonable though ("9.58 pm").
Nothing extracted yet.
Your system information
Please describe your issue in as much detail as possible:
Dates are not localized in translations. I have strings like: "VIIMEKSI PELATTU: 07/25/2018". That date should be 25.7.2018 in Finnish but instead it is in this weird American notation. I believe this used to be correct in previous versions.
Steps for reproducing this issue:
Use Finnish locale. Look at some date like last played time.