Hello PierreCA,
Global variables can be assigned with text values. Nevertheless, the variable on select won't apply a boolean logic. If you run Set(testMenu;"Some text") that will assign the text to the variable. For example, if on the same button you assign to the text property the same variable and you click it you'll be able to visualize its value in the button.
You can also validate that the action of selecting the button works by looking at the global variable.
Here is official documentation about the Set function:
Set function - Power Platform | Microsoft Learn
Now, if that isn't working it can be because the same variable was declared before with a different data type for example if you use another button Set(testMenu;1) and other button with Set(testMenu;"Some text") it will display an error as data types doesn't match Integer vs String so you need to be careful that data types match in those cases.
Hope this helps clarify the doubt, let me know in case you're still facing an issue within it.
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