One reason it might make sense to have a service like this provided by just a two or three companies in a given city instead of a large number is verification and enforcement of safety regulations such as requirements that the companies keep the scooters well maintained.
Two ways come to mind for a city to verify that a company is handling such things properly.
First, the city can have auditors regularly examine the internal operations of the company, talk to employees, and review consumer feedback.
Second, the city can pull a random sample of the company's scooters off the street and have city mechanics inspect them.
The number of scooters you have to examine from a company to determine if it is meeting maintenance requirements depends on the margin of error that you are willing to accept, not on the number of scooters in that company's fleet. Let's say you need to examine 50 scooters to get the desired margin of error.
Suppose are going to allow 5000 scooters in your city. If you do that with two companies, 2500 each, and you audit annually, you'll be doing each year two audits of the facilities and employees, and checking 100 scooters (50 from each company).
If instead you have that same 5000 scooters, but now provided by 10 companies, you'll be doing 10 facility and employee audits, and checking 500 scooters.
So same number of scooters on the street in both cases, but about 5 times the administrative costs to the city in the 10 company case is in the 2 company case.
Also, if consumers need a different app for each scooter company, a smaller number of companies will probably serve consumers better. Again, take a city with 5000 total scooters. If those were supplied be two companies, evenly split, then a consumer would only need two apps to have complete coverage. A consumer with only one of the apps would still have a decent change of being able to find a scooter in a reasonable time frame.
If that 5000 were split among 10 companies, a consumer with only one app would only have a pool of 500 scooters available. If those ended up spread out around the city it would be hard for a one app consumer to find a nearby scooter. If that company concentrated in one smaller area that could solve that problem...but then the consumer would probably end up needing the apps of several others, too, to find scooters when they are in other parts of the city.
There is going to be some optimum number of scooter companies for a given city and total fleet size which balances things. Lower numbers are more convenient and usable by consumers, and cost less to regulate and verify, but higher numbers have more competition which might lower prices.
My guess...and it is just a guess...is that three would be about right for most reasonably large cities.