It is true that, generally speaking, 融 means to melt, but has other meanings too, especially in the economic context.
In this case, as 賈可 Jacky wrote, the character 融 can be translated as circulation or to circulate. In the online Chinese Dictionary zdic this character is defined as follows:
The meaning we are interested in is #3, which says:
3. Circulation [流通]: like in the words fluctuation [融泄] (floating, drifting) and finance [金融] (circulation of money, namely savings, credit, remittances, buying and selling of stocks, and other economic activities of this kind)[...]
So, this character 融 has to do with circulation (流通) of capital (资本), and that's why the meaning of 融资 is in fact financing or funding.
This particular meaning of 融 can also be found in the 《汉语大字典》 and in the 《辞源》. Both dictionaries explain this character by saying that its meaning is basically 流通 (to circulate).
I couldn't find any reference to the ancient tradition your friend mentioned. Anyway, it seems to me that the word 融资 is quite a modern coinage.