I think you might be looking too much through contemporary eyes.

While I do not know the exact motivation behind that particular set of lyrics, the reference is probably to divorce. The child is implying they have a father and a stepfather.

The next line in the song is about a grandparent walking a child to school.

According to the AARP, 1981 was the year when the divorce rate peaked in the United States. This was the same year when Mr. Rogers first dealt with the topic. An academic paper from 1980 that reviewed the self-help literature of the late 1970s on the subject of divorce/failing relationships characterized the era as a "love crisis". This was actually based on the title of one of the books reviewed in the paper, The Love Crisis, a 1979 collaboration between journalist Carol Botwin and clinical psychologist Jerome L. Fine.

It was in this period that film and TV began to routinely deal with the subject of divorce in a serious manner. Before the 1970s, divorce was usually played for laughs (see, for examples, the film The Philadelphia Story in 1940, and The Lucy Show in the 1960s).

In the 1970s and 80s, the tone changed to being more serious, such as in films like Divorce His, Divorce Hers starring Richard Burton and Elizabeth Taylor, or Ingmar Bergman's Scenes From A Marriage.

Some of these films focused, in whole or in part, on the effect that divorce had on the children in the family. Among them are the TV films Children of Divorce starring Barbara Feldon in 1980, and Irreconcilable Differences starring Drew Barrymore in 1984. Most importantly, probably, was Kramer vs. Kramer from 1979, which won the Academy Award for Best Picture and was a box office hit.

Going back to the academic article cited above ("The Love Crisis": Couples Advice Books of the Late 1970s by Ellen Ross), the advice books available at that time gave many different perspectives on marriage. Some of the books even advocated for "open marriages", and one of them even argued that marriage should be replaced with "contract cohabitation". That is, nobody should get married, but if you do want a live-in romantic partner, it should be handled like a legal partnership, essentially via a non-marital prenuptial agreement. (Ross correctly predicted this approach wouldn't catch on.)

But the most important part of that article for the purposes of this question is how Ross concludes her piece. She actually does reference gay couples in raising children, but in a way that would back up your initial suspicion:

"To reach the millions of women and men caught in the maze of the contemporary crisis, we need to allow ourselves creatively to dream about heterosexual as well as lesbian combinations that promise security, delight, and a framework for bringing up children."

Notice how it specifically mentioned "lesbian combinations" but nothing about gay men? Even in this progressive-for-the-time article aimed at academia, the author would not go so far as to say that children might find security in a home with gay fathers. This would appear to contradict any reading of the Sesame Street lyric as referring to having two fathers who are in a romantic relationship.

Someone with more time than me could probably pull up some sources about how much gay men were villified during that era. But suffice it to say, the sentiment implied in Ross's paper was a lot more explicit elsewhere in U.S. culture at the time, that gay men did not make suitable parents.

As for divorce, Sesame Street was actually pretty late to the game when addressing the topic head-on on the show. According to an article published in Time magazine and another in the Christian Science Monitor, some of the Sesame Street staff wanted to address the topic by the late 1980s, but some of the executives vetoed the idea, arguing that their target audience of urban kids were more often from never-married couples, while divorce was more of a middle class, suburban issue.

Eventually, though, the holdouts agreed to address the topic, but as is the case with "big issue" Sesame Street episodes, the script went through an extensive review process before filming, including through their panel of child experts. The script was approved in 1992 and filmed that season. In it, Snuffalupagus's parents get divorced. But during the test screenings in front of a child audience, the scenario so upset the children that the episode never aired. It took another twenty years before Sesame Street tried again.

Nevertheless, based upon the context of the times, and the next line of the song, I am inclined to believe that the reference is to divorce, and having both a father and a stepfather, rather than the child being raised in a household by two fathers involved in a romantic relationship with each other. Maybe someone else can find info about the composition of the song in particular, but this is the most likely explanation.