Good news: This is a learned skill, not just some kind of "talent" that one is supposed to have from birth (even if it's often learned at an early age), so it's something you can work on. The short answer is you should "scaffold" it by starting easy and working up.
The fundamental issue here is about feeling the musical phenomenon that is best called "pulse"; a more common term might be "the beat," but that gets confusing in music-theory ways. The idea is that a certain point in the music is emphasized, or carries a sense of weight, and that such points recur regularly and predictably. This is "the beat" that we subdivide into rhythms. We should note that not all music is "pulsed". Most is, and has been across history and many music-cultures, but some music can be unpulsed. And some music can prefer a "weak" pulse, that is not strongly emphasized, or to be flexible with it (aka "rubato"), stretching or compressing the period between pulses expressively. Other music might change the pulse often (changes in tempo), or have a complex meter that combines pulses of different lengths.
My advice is to avoid these at first: start with something "strongly pulsed," with a straightforward meter. There's a reason that the notion of "danceability" is correlated with "strongly pulsed." A lot of funk or disco can work well here: tracks with strong, thumpy, straightforward four-on-the-floor bass lines.
When I work on this with students, I often play this track in which RJD2 remixes Astrud Gilberto:
The opening 14 seconds or so are unpulsed, but then a bass-range piano part comes stomping in, and starts strongly marking every downbeat. This is maybe not the simplest track to choose, because the snare drum also provides a strong "backbeat"; that is, if the low piano notes are beat 1, then the snare plays on the so-called weak beats 2 and 4. The bass drum also complicates things a bit. But overall, with the help of those left-hand piano downbeats, the pulse is marked so strongly that it's easy to move in time with it. I encourage students simply to bend their knees with the pulse.
Continue with this kind of exercise, simply locating "the beat" in strongly pulsed music. "Stay alive" with the Bee Gees, maybe. Dance with the "Dancing Queen." Walk through the "Gangsta's Paradise." March along with John Philip Sousa. Just keep bobbing in place, bending your knees. Try replaying a small handful of tracks often; the ability to predict the familiar is easier than the ability to pick up and react to something new.
Once this ability becomes so secure that it's boring, you can move on to examples that are more subtle about their pulse, not thumping it out on every beat. You can try another meter, like a waltz, so that the pulses come in sets of 3 instead of 4:
Then you might try tracks that change tempo, and try to follow along. Grieg's "In the Hall of the Mountain King" speeds up as it goes:
For the ultimate challenge, try the sort of pieces that manipulate the pulse so much that it approaches being unpulsed; try some Chopin piano pieces (famous for extreme "rubato," or pushing/pulling the pulse) or Impressionists like Debussy or Ravel. A piece like the Debussy violin sonata might present the ultimate challenge, with the tempo constantly in flux; I've told students before that no two consecutive beats should be the same length.
That piece not only manipulates, but also de-emphasizes the pulse, "smoothing it over" with ties and blurring its impact. A choreography of it might be equally fluid and expressive.
Edit after question was edited: Looks like the issue is not just finding beats, but finding phrase lengths. It gets a little confusing, since dancers often count the total number of beats in a phrase, while musicians describe it as a number of measures (there's a reason dancers often count off "five, six, seven, eight" while musicians say "an' a one an' a two an' a one two three four"). For being aware of phrase lengths, it might actually help to do some study of music theory, enough to know a "cadence" when you hear it, and especially the part that deals with deciding just how long phrases are. This is no small undertaking; we're talking about a semester or two of an average theory course, but it would be well worthwhile.
Meanwhile, the advice is the same: start simple. Stick to songs that do have simple phrases, in sets of four bars, and avoid things that throw in an extra bar or two now and then (like Simon and Garfunkel's "Scarborough Fair"). And work with the same tracks over and over so you get to know their ins and outs.