Can a button know in what row he is?

Yes, you can do that with scripting. The sample script called “Record picker” (accessible when you create a new Scripting app) does mostly what you’re requesting. It’s just a matter of changing the field that you want to display once the record is selected. See below:

// Change this name to use a different table
let table = base.getTable("Button Test");

// Prompt the user to pick a record 
// If this script is run from a button field, this will use the button's record instead.
let record = await input.recordAsync('Select a record to use', table);

if (record) {
    // Customize this section to handle the selected record
    // You can use record.getCellValue("Field name") to access
    // cell values from the record
    output.text(`You selected this record: ${record.name} (${record.getCellValue('Record ID')})`);
} else {
    output.text('No record was selected');
}