Merc will insist it's badged Mercedes Jeri, have no doubt of that.
They already power both McLaren & Aston Martin so it's not really unusual for them to supply other manufacturers, maybe these others are not the size of Renault but does it really make a difference?
The cost cap model for F1 makes it more profitable, well less likely to lose much money anyway, to be an engine purchaser than an engine supplier. 
Aston Martin, the car company, uses Mercedes engines in their road cars, so it's not really a big deal for them to use the Mercedes badged engines. McLaren has a history of success running as McLaren-Mercedes, I think they were trying to bring back good vibes, as much as anything else.
Other teams that ran Mercedes engines were privateer teams that I think gained a little status running Mercedes engines, especially in the early turbo hybrid era.
Alpine is in a different situation than anyone else running Mercedes engines. I don't know what the rules are about supplying engines in terms of branding. I'm pretty sure there's a fixed cost to use an engine supply (I don't know if it's per unit, or on a season long contract basis, I don't really care), and suppliers are required to be able to supply X number of teams (again, I don't really remember how many X is). Mercedes may not get a say in what the name of the Alpine power unit is, as long as Alpine and Mercedes are upholding their ends of the deal.
Given that there are murmurs of Mercedes being 'embarrassed' to be outscored by a customer team in McLaren, maybe they aren't keen on being potentially outscored by two customer teams. Sure, lots of us will know that the Alpine BWT engines are really Mercedes, but I don't think our opinion on Mercedes losing to a customer is as important to Merc.