Search "private label pet clothing" and most results use the term loosely | sometimes meaning a generic product with your logo added, sometimes meaning genuine customization, sometimes meaning full custom design. The vagueness isn't usually dishonest, but it leaves brands picking a manufacturing path without understanding what they're actually committing to in terms of cost, timeline, and how differentiated their product will really be.
The three models | white label, private label, and OEM | sit on a clear spectrum of control versus speed. Picking the wrong one for your stage costs either money (overpaying for OEM customization you don't need yet) or differentiation (settling for white label when your brand needs more distinction to compete).