Update: A generalization to translate months and days:
ClearAll[translateDates, translatedDateTicks]
translateDates[f_: (Capitalize[StringTake[#, 3]] &)] := StringReplace[
CanonicalName[#] | Capitalize[StringTake[CanonicalName[#], 3]] -> f[#2] & @@@
Join @@ LanguageData[#, {"MonthNames", "Days"}]] &;
translatedDateTicks[f_: (Capitalize[StringTake[#, 3]] &)][l_String,
n_: All, ds_: "MonthName"] := Replace[System`DateListPlotDump`DateTicks[{##}, n, ds],
s_String :> translateDates[f][l]@s, All] &;
Examples:
data = {#, RandomReal[]} & /@ DateRange[{2020, 1, 1}, {2020, 02, 15}, "Day"];
DateListPlot[data, ImageSize -> 800, LabelStyle -> 12,
FrameTicks -> {{Automatic, All},
{translatedDateTicks[]["Portuguese", 30, "DayName"],
translatedDateTicks[]["Spanish", 30,
{"DayName", "\n", "MonthName", "\n", "YearShort"}]}}]

data2 = {#, RandomReal[]} & /@ DateRange[{2020, 1, 1}, {2021, 01, 31}, "Month"];
DateListPlot[data2, ImageSize -> 800, LabelStyle -> 12,
FrameTicks -> {{Automatic, All},
{translatedDateTicks[Identity]["Chinese", 15, "MonthName"],
translatedDateTicks[Identity]["Russian", 10, {"MonthName", " / ", "YearShort"}]}}]

Original answer:
ClearAll[monthShort, dTicks]
monthShort[lang_] := CanonicalName[#] -> Capitalize[StringTake[#2, 3]] & @@@
LanguageData[lang, "MonthNames"]
dTicks[l_String, n_:All] := System`DateListPlotDump`DateTicks[{##}, n, "MonthName"] /.
monthShort[l] &;
dTicks[a_Association, n_: All] := System`DateListPlotDump`DateTicks[{##}, n,
"MonthNameShort"] /. a &;
Example:
monthNameRule = <|"Jan" -> "Jan", "Feb" -> "Fev", "Mar" -> "Mar",
"Apr" -> "Abr", "May" -> "Mai", "Jun" -> "Jun", "Jul" -> "Jul",
"Aug" -> "Ago", "Sep" -> "Set", "Oct" -> "Out", "Nov" -> "Nov", "Dec" -> "Dez"|>;
DateListPlot[{#, RandomReal[]} & /@ DateRange[{2020, 1, 1}, {2021, 01, 31}, "Month"],
ImageSize -> Large, LabelStyle -> 16,
FrameTicks -> {{Automatic, All}, {dTicks[monthNameRule, 10], dTicks["Portuguese"]}}]
