I don't know how/whether the WolframCloud determines your local timezone, but my desktop version always has "knowledge" of my local timezone (there are things I can do to override that knowledge, but that's not really relevant). The dates presented to me, unless I specify a timezone, are always presented as the time in my local timezone. So, notice that in your case, the function that you're mapping has no timezone specified. So, the results are shown to you as if they were in your local timezone. The third date object was a time specified with the local offset of Helsinki, which would be several hours different from wherever you are, apparently. So, the date presented to you, which again is local to you, has been shifted to account for this.

It seems that just by looking DateObject itself things are correct in Wolfram Cloud:

{$TimeZone, DateObject[], DateObject[TimeZone -> None], 
DateObject[TimeZone -> "Europe/Helsinki"]}

{3.,
 DateObject[{2025, 8, 18, 20, 52, 41.576097}, "Instant", 
  "Gregorian", 3.], 
 DateObject[{2025, 8, 18, 20, 52, 41.576202}, "Instant", "Gregorian", 
  None],
 DateObject[{2025, 8, 18, 20, 52, 42.353321}, "Instant", 
  "Gregorian", "Europe/Helsinki"]} 

but I I pass DateObject[] and DateObject[TimeZone->"Europe/Helsinki"] to DateString something goes wrong:

DateString[#,<|"Elements"->"LocaleDateTimeShort"|>]& /@ {
  DateObject[],
  DateObject[TimeZone -> "Europe/Helsinki"]
}

{"Aug 18, 2025, 9:02:36 PM", "Aug 19, 2025, 5:02:36 AM"}  

Even the day is wrong for the DateObject with TimeZone. There is no such error in desktop version. If I define Language and remove Elements specifications:

DateString[#, <|"Language"->"Finnish"|>]& /@ {
  DateObject[],
  DateObject[TimeZone -> "Europe/Helsinki"]
}  

{"18.8.2025 9:07 ip.", "19.8.2025 5:07 ap."} 

So time is again wrong.

However if I remove Elements and Language specifications, thinks starts to look OK:

DateString[#] & /@ {
  DateObject[],
  DateObject[TimeZone -> "Europe/Helsinki"]
}
{Mon 18 Aug 2025 21:06:19, 18.8.2025 9:06 pm} 

So looks like that Element->"LocaleDateTimeShort" and Language specifications somehow break the textual presentation of DateObject.

In the meantime, you can force the correct result by adding an explicit TimeZone -> "Europe/Helsinki" option value to the DateString call itself:

In[102]:= $TimeZone = 3;
DateString[#, <|"Elements" -> "LocaleDateTimeShort"|>, 
   TimeZone -> $TimeZone] & /@ {DateObject[], 
  DateObject[TimeZone -> "Europe/Helsinki"]}

Out[102]= {"Aug 18, 2025, 10:02:23 PM", "Aug 18, 2025, 10:02:23 PM"}