Although HSBC operate in both countries, they are getting out of retail banking in Canada, and have quite a small presence in France (mainly the old CCF branches in the Paris area). I'm not aware of other banks that operate retail banking in both countries.
There is no reason why you should have the same bank in each country, and plenty of reasons why you shouldn't. If one bank goes bust, or has computer problems, or freezes your account by mistake, you have an alternate with debit cards to use.
You should be able to mostly avoid bank fees and charges in both countries. La Banque Postale charges a few Euros a year. Most Canadian banks have an account where you pay no fees when you keep a minimum balance of a thousand or two CAD.
If you had one bank, you would have to use their currency exchange rates. You can almost always do better elsewhere.
For conversions from EUR to CAD, it's hard to beat Currencyfair. Pay the money to them from your French bank with bill pay, exchange at peer-to-peer rates that can be close to the interbank rate, and transfer out to your Canadian account for 4 CAD. The better exchange rate easily covers this fee.
For CAD to EUR, it's not quite so easy (as you can't bill-pay into Currencyfair in Canada), but many use xe.com, which can be paid by bill-pay from some Canadian banks. Not quite such good rates as Currencyfair, but still better than any bank.