It was not particularly common, but it would depend a lot on the religious beliefs of the parents. I don't know anything about Walt Whitman's brother specifically, but the likely reason for the child not having a name was that the child was not baptized.

Back then, a child "officially" received its name, as far as officiality of these things went back then since there was no civil birth registration, by being baptized/christened in the church. A child could not be baptized/christened without being given a name. The priest/minister recorded the name in the parish baptismal record book, and those books were sent periodically (usually yearly) to the church authorities--to the Vatican in the case of Catholics, to Canterbury or Westminster in the case of Anglicans, to a counsel of bishops who gathered at the annual synod in the Presbyterian and Reformed churches, etc. In Congregationalist churches with no affiliation with a national or international body, they were kept there at the local church, but the records were kept.

Typically, people were rather eager to have their children baptized precisely because child mortality was so frequent. The baptism usually took place the next time after the child was born that the mother and child were healthy enough to make the trip to church, so for people who lived in town, it might only be a week or a few days, while for people who lived in the countryside further away from the church it might not be for a month or more, depending on what time of year it was and what had to be done on the family farm before they could take the time off to get to the church.

There were exceptions to this, of course. Some people didn't attend church regularly, so they might wait until Easter or Christmas to have their child baptized. And there were some people who didn't attend church at all. And then there were some Christian denominations that didn't believe in baptism.

Which brings us back to Whitman: he was raised in the Quaker tradition (not "church" because Quakers don't believe in "church" per se). And instead of baptisms or christenings with sacramental rites, Quaker children go through a "nomination" process where the child is presented to their Friends with a name, and Friends of the parents "witness" the naming and that becomes the child's name, though it's not written down in a book anywhere (because that would be too pompous). There wasn't any real ceremony to it like in other Christian denominations.

And due to the theology of Quakerism, there was frequently less social pressure to be a regular meeting-goer, or to have the "nomination" be solemnized immediately after birth than there would be with other Christian sects in America. A lot of Quaker society was based around visiting each other at home (1, 2), rather than being focused on rituals and preaching at the Meeting House. (In smaller Quaker communities, the homes of your friends and neighbors were your Meeting Houses, after all.) Quakers don't believe in ministers, or tithes, or sacraments, either. You showed up when you showed up and when you did, you had your kids nominated to be part of your "society" of "friends".

So, while I can't speak of Whitman's brother's specific circumstances, this would be one of the primary reasons for the child not being named: the fact that the child wasn't baptized or christened into a church, or nominated into the Society of Friends. It wasn't that the parents didn't call the baby by any specific name for six months, though that might be true. It's that they never got around to going to the Meeting House and nominating the child as a member of the Society of Friends, so when the child died, he had never been given a name accepted by the Friends of the family, so his parents didn't feel right burying the child with a name on his tombstone. And nominating the child after death was out of the question (posthumous baptisms were more the purview of Mormons later on).

Because social life revolved so much around religion in those days, it's important to remember that the child not having a name wasn't an indication that the parents were calling the child "baby" and nothing more for six months, though it is possible. It's more likely an indication that whatever they were calling the baby, they hadn't had the chance or hadn't bothered to make that name official with the religious body the parents belonged to.

All this, of course, is specific to the Christian religious history of America, where Whitman's family lived. This all applies to Christianity in Europe at the time as well, though the denominational mix there was different. As far as other religions in America at the time, such as Judaism, I can't really say, nor can I really say anything about what was happening outside of America and Europe, either.

TL;DR: Before the rise of civil birth records in the second half of the 19th Century, any child that died before the child was baptized, christened, or nominated into a religious sect was typically buried without a name on their gravestone. That doesn't mean their parents didn't privately call the child by some name before the child died, but since the child had no baptismal name, it wasn't seen as "official" in their eyes, because it hadn't been accepted by the church, and, thereby, it hadn't been accepted by God, so the child died "nameless".

Here is an article published around the time of Walt Whitman's youth from the Friends Intelligencer explaining why the Friends (Quakers) didn't participate in water baptisms of children, with some details about the nomination process.

Here is a book that talks about the lack of Quaker rituals and record-keeping, including of births and baptisms.

A section in this book about the population history of England singles out the Quakers for being at odds with other Christian sects when keeping track of populations due to their lax approach to birth record keeping.

This book details the history of civil birth registration legislation in England, which at first was just a system of the various churches sending their baptism records to the government, but also required Quakers to send in a list of children solemnized in their Society even though they weren't in the habit of keeping baptismal records.

More comprehensively, you might want to look through The Oxford Handbook of Quaker Studies. More generally, you may be interested in Children in Colonial America ed. by James Marten which includes some details of child baptisms in various American Christian denominations, as well as Quaker social customs, in early America.

FURTHER READING:

Friends and Neighbors: Group Life in America's First Plural Society by Michael Zuckerman

The Quaker Family in Colonial America: A Portrait of the Society of Friends by William J. Frost

Quakers and the American Family: British Settlement in the Delaware Valley by Barry Levy

EDIT:

And the article that /u/augmentedseventh linked to also has good information:

"Infant Mortality and Child-Naming: A Genealogical Exploration of American Trends", The Journal of Professional and Public Sociology, by Al McCormack