While many of the characters from the television series were based on the real life characters as Laura Ingalls Wilder wrote about in the Little House books, Albert Ingalls was never a character in the books, nor did the real life Charles Ingalls ever adopt a son, either officially or unofficially.
Albert Ingalls was also not based on any known living person, but rather created by the series writers when they wanted Charles Ingalls to have a son.
The television took much license with the characters of the show, particularly after the first season. So there are many characters who never existed in real life nor in the books, such as Mary’s husband Adam Kendall on the television show, when the real life Mary Ingalls never married.