French citizen visiting Canada through USA

As a French citizen you can be admitted to Canada without a visa as a visitor for either tourism or business. Going to a job interview is an allowed activity. When you show up at the border the officer decides how long you will be allowed to stay, based in part on your reason for bring there. That can be for up to six months but may be shorter.

When you leave Canada and attempt to reenter you have to go through the process again. You explain your reason for being there, and the officer decides how long to admit you for. If you have recently been in Canada it will likely be less than six months. If you do it too frequently or for too much time you may be suspected of trying to live in Canada through repeated visits, in which case you would be denied entry.

Which brings us to your reason for being in Canada. Six months is a long time to spend job hunting, so I have to point out that you cannot work in Canada without a permit, for any length of time at all. You or your employer would need to apply for a work permit once you had found a job. Failing to do that can get you deported and banned. Once you have the work permit a six month limit would not be a problem, nor would reentry.

As a visa waiver program visitor in the US, you are limited to a 90-day stay. There is a somewhat poorly conceived rule that is designed to prevent people from making short trips to Canada to get around the 90-day limit, so if the total time you'll spend in both countries is longer than 90 days then you might indeed run into trouble trying to get into the US. I suppose that would be a reason for someone to say that your stay would be limited to three months.

One thing to note about that rule is that it's discretionary. If you're spending more time in Canada then than the US then you might have a good chance of the CBP officer letting you back into the US even if you would run afoul of the rule.

Officer at the boarder will decide how long you can stay really. I have entered Canada multiple times and it is totally up to the officer. Based on average of 5 entries. 4 out of 5 I have been granted entry for 6 months.

The one time I was granted exactly 2 months wasn't that big of a deal really. Since you can always apply for extension online. It was the officer inexperienced with my passport (Officer: First time I see a passport like that).

Applying to extend your stay online can take up to 4 months sometimes. But during that time you are considered an implied status. Implied status means you should stay in Canada. If you leave then your extension application will be cancelled. You should have an address that they can mail your visit extension permit to. You should prove having enough funds to support your extended stay.

More on how to apply to extend your stay is at https://www.canada.ca/en/immigration-refugees-citizenship/services/visit-canada/extend-stay.html