This is a real chicken-and-egg problem. However, the thing that makes the question answerable is that time signatures are placed as the starting point.
Time signatures are, after a fashion, a declaration of convention. A time signature of 2/4 is like saying "there will be emphases of roughly equal value every other beat." 4/4 is like saying "there will be emphases every two beats, but the first of them will generally be stronger than the second." 6/8 is like saying "I want two 'large' beats per measure, in a strong — less-strong pattern, with each 'large' beat subdivided into three smaller beats."
[In 4/4] what, exactly, is making beats 1 and 3 stronger?: Convention. And, of course, we see jazz and pop routinely play against this convention by placing emphasis on beats 2 and 4, the back-beat.
Is the composer doing something that makes us feel a strong beat, similar to how poets carefully choose words with stressed syllables in particular positions for different types of meter? Yes. The composer chooses a time signature not unlike the way a poet chooses, say, a limerick, haiku, or iambic pentameter.
(Are composers typically taught to do something to emphasize strong beats in a piece?) Yes. Leaving aside those who have a natural intuition for meter, meter and its conventions are routine part of elementary instruction for any musician, composer or performer — and this includes musical cultures whose metric structures are not based on the same comparatively rigid principles of Western music.
So is it something mostly psychological perceived by listeners due to certain expectations? Up to a point. Again leave aside those with a natural intuition, many people aren't consciously aware of meter, if aware at all — hence its being part of elementary instruction, and not infrequently a particularly difficult part.
But once one has learned or intuited the musical language, then there is an expectation — one intentionally upset by composers attempting to garner some musical impact, as mentioned in relation to the back-beat in jazz and popular musics.
Do we ... necessarily perceive or feel a certain time signature or meter, absent non-metrical accents? No. If metrical accents were inevitable, then a great deal of music could not exist: unmetered chant, for example.
The heart of the question
Metrical accents, more often than not, are achieved via dynamic emphasis — usually, but not necessarily, subtle emphasis. They can be achieved in other ways as well, but what makes them metrical is their regularity in time.
What differentiates a dynamic accent, even if it corresponds to a metrical emphasis point, is that it's given additional emphasis, beyond what would be expected from the meter alone. At this point, though, we are solidly in the realm of perception and aesthetics, since the question of how loud a metrical accent should be and how much louder a dynamic accent should be are not rigidly defined.
Consider speech. Certain syllables of certain English words are emphasized — typically by way of a "normal" accent, an expected level of emphasis, but a level unique to individual speakers, subjects, situations, etc. But a dynamic or agogic (or both) accent might be added to an already accented syllable for even more emphasis. And this same principle can be applied to (metered) poetry: there's the expected, metrical syllabic stresses, and the additional stresses added by the reader.