Can the Philanthropist use their alien power only once, giving away one card, or can you continue to give away cards in one encounter?

Just one.  (Most effects work this way, but there are a few exceptions which I'll list below.)

From the very beginning, Cosmic Encounter has never been real precise about explicitly stating whether "one" means "once per encounter" or "once per use of this effect", so you have to let context be your guide.  In this case, if the power could be re-used repeatedly then there wouldn't have been any point in saying "one" in the first place ... in other words, if the intention was multiple cards it would have said something like "give away one or more cards" or "give away cards one at a time".  Sometimes it's helpful to work it backwards ("if they meant A, would they have written it this way? what if they meant B?") to infer the intent.

As a general rule, if "one" is used in the context of a specific event that could occur more than once per encounter, then you can do that "one" thing more than once per encounter (each time the event arises). But these are rare; most of the time, "one" is used in the context of something that cannot occur more than once per encounter, or is linked to a particular Phase or an element of a phase, such as inviting allies, forming alliances, revealing cards, winning, losing, dealing, etc.

So most things (the overwhelming majority) that say "do one x" are once per encounter. I could only find a few exceptions:

• Fury places "one" token on her sheet every time she loses one or more ships.  If she loses ships for three different reasons during an encounter, that's three tokens.
• Healer can heal multiple players per encounter (or even the same player multiple times), drawing "one" card each time.
• Remora gets "one" ship or card every time somebody else gets ship(s) or card(s), respectively.
• If three destiny cards are flipped in a row, Shadow executes three ships.
• There are a very few flares (such as Wild Zombie) that could in theory let you do "one" of something more than once per encounter, but only if you were using a house rule of totally unrestricted flares (allowing re-use of the same flare in the same encounter).

These few exceptions all clearly place the "one" thing in the context of some particular repeatable event.  Pretty much everything else is in the context of a particular Phase or a once-per-encounter occurrence.  The Philanthropist (to return to your specific question) operates in the Alliance phase "after alliances are formed", a timing window that occurs only once per encounter.

Always seemed pretty clear to me.  Says you can "give one card," not as many as possible.  Remora, Healer, etc. make it clear their powers activate per event.  Philanthropist doesn't, so you could think of the triggering event as the encounter phase it activates in.