Bike batteries - replace 36V /10Ah battery with a 36V / 13Ah?

Think of the voltage as the "type" of fuel for your car: Regular, Diesel, etc. You don't want to mix those. But the Ah rating is more like the size of your gas tank, a bigger one just lets you go farther between refueling.
There is one other item to consider, which I think is unlikely to be an issue here as we're talking 36v batteries.

Every ebike battery has (i hope!) a BMS - a Battery Management System. Your motor has a controller... think of the BMS as the controller for the battery, which if you take it apart you will find it is not one battery, but dozens of little ones more or less welded together to make a single whole. The BMS manages all of these cells that make up your battery pack.

I won't go into all the things the BMS does, but the relevant one here is the 'peak' current draw that the BMS allows the motor to ask for. Lets say for the sake of argument your BMS has a peak current setting of 30 amps. If your motor asks for 35 amps, the BMS will sense this and shut off the battery to protect it. To restart it again, you will have to plug the battery into a charger... so just turning the bike off and then on again won't do it, and you'll have to ride the bike home without power to get it going again.

So the other thing you need to pay attention to is not just the battery plugs, and the voltage, but also the BMS' peak power setting (there is also a continuous power setting but lets just stick to talking about the peak).

Now, I think its pretty likely @darf will not have to worry about this. Any bike with a 36v system is unlikely to need enough power to trip a BMS. Very likely any 36v battery will have enough peak current capability to run what is likely to be a fairly basic, low-ish power hub motor. But its never a bad idea to cover this base and know for sure what the battery's current limits are. Once you buy the pack thats a few hundred bucks gone forever so you don't want any surprises.