Is it customary for people in Portugal, like in Spain, to assume both their fathers' and mothers' surnames?

Yes, I have one surname from my mother and two from my father.
Unlike in Spain, where the paternal surname comes first, in Portugal it used to come in the end.
Nowadays it is indifferent and you can juggle as you like it - f.i. father-mother, mother-father-mother; father-mother-father or even find a surname from an ancestor if you prove you are related to him at a relatively close degree.

I know that a person transmits only his father's surname to his children. So this gets me wondering about dynastic names: both Queen Maria II and her niece, Princess Imperial Isabel of Brazil, were members of the house of Braganza. Accordingly, their children would likewise have all been Braganzas as well.

Not exactly.
Although they would generally be known as the House of Bragança, some historians consider the descendants of D.Maria II as a second house of Bragança, the house of Bragança-Saxe-Coburg-Gotha and sometimes you will see them referred as f.i. D.Carlos de Bragança-Saxe-Coburgo-Gotha.

But what of their grandchildren? Can a son who inherits his mother's surname transmit this to his own children? As I understand, Isabel still has male-line descendants today: of course, they're all members of the house of Orléans, since they all claim King Louis-Philippe of the French as a patrilineal ancestor.

They are just members of the Brazilian Imperial House of Orléans-Bragança, not of the French RH.
When the monarchy was abolished in Brazil and the Count of Eu wanted to be fully reintegrated in the House of Orléans with rights of succession, his pretensions were refused - see the Family Pact of 1909.

However, I've noticed that they're all styled with the hyphentated name Orléans-Braganza: does this mean that despite acquiring a surname through a female, it can nevertheless continue in perpetuity through the male line of descent?

I believe that the male-line grandchildren and great-grandchildren of Queen Maria II of Portugal were also recognized as Braganzas (although her descent in the male line eventually became extinct).

In Spain, the reason that the Bourbon house name got preserved was that Queen Isabel II married an agnate -- her first cousin, Infante Francisco. But her Portuguese counterpart married (as her second husband) Prince Ferdinand of Saxe-Coburg and Gotha -- not her uncle Miguel (as had been expected).

The house name was preserved but you had Don Juan de Borbon y Battenberg, D.Juan Carlos de Borbon y Borbon, D.Felipe de Borbon y Grecia and Pss Leonor de Borbon y Ortiz.
Probably, if Leonor will reign one day and have kids, her children will have Borbon as first surname as if she was a man, and her husband will have to settle with the second surname or may be they would just abolish the second surname like in other RH's