Many people believe that the Mourner’s Kaddish is the Kaddish, but although it is important, there are four other forms of Kaddish. So we say various forms of Kaddish regardless of whether we are mourning.
But looking more closely at the forms of Kaddish, we see that the Kaddish DeRabbanan (“our rabbis’ kaddish,” recited after studying sacred texts) is said by anybody, and it includes the entirety of the Mourner’s Kaddish, so we might read from this that it is permissible for anybody to recite the Mourners’ Kaddish as well.
In some congregations nowadays, the whole congregation stands and recites the Mourner’s Kaddish in unison, as a sign of support for those in mourning, or for those who may not be as fluent with the recitation and thus suffer embarrassment.
We do not all stand together here at Beth Shalom, but we do not discourage it. And if you wish to stand and join in Mourner’s Kaddish although you are not directly a mourner, you absolutely may do so, either as a sign of support or for any of the millions who may not have anyone to say Kaddish for them.
Some additionally ask whether they should rise at a shiv’ah minyan and say Kaddish with the family if they themselves are in mourning - within sheloshim (the first 30 days of mourning) or during the first year or on the yahrzeit (anniversary) of a death. Yes, it is perfectly acceptable and even encouraged that mourners support one another, even at a shiv’ah for one or the other.