You need to use the <<print>> macro to output the results of your either() function calls.
Your second <<if>> has a improper conditional, the logical-OR operator joins sub-expressions, so you must provide complete expressions on both sides. That said, you're attempting to check the contents of an array there and in the following <<elseif>>, which you do with one of the includes family of array methods—i.e. <Array>.includes(), <Array>.includesAll(), <Array>.includesAny()—rather than with any of the standard equality or relation operators. In this case, you probably want <Array>.includesAny() for the <<if>> and simply <Array>.includes() for the <<elseif>>.
For example:
<<widget "ass_aya">>\
<<if $ass_size is 1>>\
<<if $ass_size_1_titles.includesAny("маленьк", "тощ")>>\
<<print either($ass_size_1_titles)>>ая\
<<print either($ass_titles)>>а\
<<elseif $ass_size_1_titles.includes("ужат")>>\
<<print either($ass_size_1_titles)>>оя\
<<print either($ass_titles)>>а\
<</if>>\
..................
<</if>>\
<</widget>>
Additionally. If the $ass_size_1_titles array isn't going to change, you could actually use an <<else>> in place of the <<elseif>>, since if the <<if>> was not true then the <<elseif>>'s conditional will always be true. In other words, with the $ass_size_1_titles array as-is, you could probably get away with the following:
<<if $ass_size_1_titles.includesAny("маленьк", "тощ")>>\
<<print either($ass_size_1_titles)>>ая\
<<print either($ass_titles)>>а\
<<else>>\
<<print either($ass_size_1_titles)>>оя\
<<print either($ass_titles)>>а\
<</if>>\
PS: In the future, please use the code snippet widget—it's the first item on the editor bar—when posting code or markup.