Does the word "pleased" work in this sentence?

When she retired from the job, the manager pleased Jane with a symbolic gift.

No.

You could possibly say "The manager made Jane happy by giving her a farewell present".

It's also potentially ambiguous. Who does "she" refer to, the manager or Jane?

Only potentially. Regular practice tells you what's meant. It's much more common for the person leaving to receive a gift than for them to give one.

Barque said:

Only potentially. Regular practice tells you what's meant. It's much more common for the person leaving to receive a gift than for them to give one.

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It's ambiguous, nonetheless. It's clearer to say "When Jane retired, her manager gave her a lovely [leaving] present".

We can't assume that every culture is the same. I learnt that quite fast when I taught in Madrid. I found out in advance that one of my students had a birthday on a day we had a lesson. As would be the done thing in the UK, I bought her a birthday card and a small cake on the way to the lesson. When I arrived, I was very surprised to see a box of cakes already on her desk. I assumed her colleagues had bought them for her. However, she then told me that in Spain the person celebrating their birthday buys cakes for everyone else! Also, they don't do birthday cards (which explained why it was so hard for me to find one!)