Does it matter where my furnace condensate pump gets power from?
Are there any other connectors (not power)? If these are all connected up then I wouldn't have thought it matters.

The only possible thing I can think of is that if the condensate pump power is turned off and the furnace keeps running it could cause problems - but I assume that there are alarms/indicators for this (how did you know the original failed?).

It might look neater if you do use the furnace's electrical panel - assuming it delivers enough power of course.