Avalanche is not connected to Binance Smart Chain (BSC) network, although addresses look the same on both networks. If you do a withdrawal from Binance using BSC network (as BEP20 tokens), funds will actually not arrive to Avalanche. Instead, they will be sent to BSC network address you entered, and private keys for that address will be necessary to move them.
As Binance uses the same addressing scheme as Avalanche, private keys that control the address on one network will work on the other too. So, if you tried to send to address on Avalanche that you control, your private keys for that address will work on BSC too. Using Metamask with the Avalanche account, you need to switch to BSC network, as described here. After switching you should see the same address as on Avalanche.
You probably won't see your funds yet, as they need to be added as custom token. The address of AVAX BEP20 tokens on BSC is 0x1ce0c2827e2ef14d5c4f29a091d735a204794041. Use that address to add AVAX as a custom token. Follow the instructions from "How to add a custom token?" heading.
After adding AVAX token, you should see your funds. You can now deposit them back to Binance, and follow the instructions on how to withdraw AVAX to Metamask.