Im wondering if he didn’t really marry this second ‘wife’ and her younger children were not really his, just using his name.
Im reasonably confident it’s him because 2 of his children and also in the household
It may just be the way the census was filled in, or they gave the information to the enumerator. You have the head of the household, if that's a man then his wife, then the other family occupants usually in descending order of age, then any boarders or servants.
I have seen several instances where the same surname is used for head, wife and their joint & collective children even though some are step children. (And a few instances where step children are described as "in laws".)
And in a time when divorce was beyond the reach of ordinary folk, it is very common for the woman to use her second 'husband's' last name in the census and presumably in her daily life but revert to her legal last name for death certificate, will, gravestone.
Conversely their children will likely be registered with their mother's legal last name (even if their father is on their birth certificate) but use their father's last name in daily life and continue to use that last name for their own marriage, death etc.
So to know whether the children are his you need to track them into their adult lives to see what names they use, hopefully find a marriage record to see who they name as father and or order their birth record.