Can I receive BTC from an exchange (exchange ‘A’) with electrum wallet, and send the BTC to another exchange (exchange ‘B’)?
. . or does it have to be used with the same exchange where the BTC was bought?
You can also deposit on Exchange B from Exchange A using the deposit address of Exchange B as the withdrawal address of Exchange A when you cash out from there. But I would not suggest you to do this because there are some risks involved in this. It is always safer to first withdraw to the personal wallet and then deposit on the next exchange.
Like OSS said, bitcoins are not specific to a exchange and like DWM14 said, be careful about the SegWit/Non-SegWit wallet address and check which one is supported by Exchange B first before sending.