While "senior," originally "senior" citizen, seems to be a kind of modern smarmy euphemism—there's a perhaps apocryphal story about the gentleman who protests, "I'm not a 'senior citizen,' I'm an old man!"—it is in fact the Latin word for "older." It has for some time meant "having longer tenure in a rank," as in "senior officer." In that sense, it replaced "elder," which was used for that sense of "senior" in the mid-18th century. So the extension "senior citizen," i.e., someone who has been a citizen longer than a "junior citizen"* is logical. "Senior" alone is used by those for whom the extra three syllables (or extra space and seven letters) of "senior citizen" are just too, too exhausting or time-consuming.
*A phrase I have never encountered to describe someone who isn't old enough to be a "senior citizen." But it would be logical antonym.
In the US, the term is currently understood to mean someone who has reached the traditional retirement age or the age at which they may received Social Security payments, in both cases 65 years. That will likely be a higher number in the not-too-distant future, as it was established when life expectancy was much lower than it is now.
Yes, you can refer to the older people in the community as 'seniors' but why not simply call them 'older people'? Almost all, if not all, the other terms are unacceptable for some reason to some section of the people concerned. The so -called synonyms like ''seniors' or 'the elderly' or, god forbid "old age pensioners", yuck and ugh, carry all sorts of connotations that risk being unacceptable.
I've heard BE speakers say "He's an OAP", "He's retired" and "He's a pensioner", but I have never heard a BE speaker say "He's a senior". BE possesses the term "senior citizen", but it tends to be confined to writing. So - I cannot imagine myself saying "There are some dignified seniors in our building".