I have two responses...
The immediate, practical comment is that you can join the person and household records by st and serialno to get person weight attached to the housing record. (Note: you'd also need to add additional processing to make sure that you're counting housing units only once, because joining housing and person records will "expand" the housing records for each person in a household.)
The bigger picture response requires more discussion.
Could you please provide a bit more context about your goal with this analysis? What are you trying to accomplish by defining housing and group quarters "types?"
The reason I ask is that, to me (at first glance), combining housing unit and group quarters records seems a bit like mixing apples and oranges. Housing unit records have a clear, countable definition. Group quarters records do not, like housing units, reflect a "building," or an "institution," or even a capacity count. They (if my memory serves) act more like a placeholder for a person record. Group quarters (population) counts can rise, or fall, within an institution simply by people moving into (or out of) the facility. So even appending the person weight will only give you the number of residents represented by that record, in that GQ facility, for a given point in time. It does not reflect facility capacity. (There have also, historically, been issues with respect to the weighting, by GQ type, at a sub-state level... but that's another thread for another time.)
So if you're trying to compare housing inventory with a similar measure of group quarters inventory (number of beds, number of spaces available, etc...) you might want to take these limitations into account.