Even if Scribus allowed same names of text frames, this wouldn't solve anything. What you need here is JavaScript. Here's an example what you can do (note: I'm not acquainted with JS, so there might be an easier and more elegant solution than this one I'm presenting here):
1. Create all necessary text fields, but for boxes where one should enter data don't use "Insert Text Frame", but "Insert PDF Text Frame" instead. So, e.g., you'll have on the left a simple text frame containing "First Name:" entry, and on the right should be a placeholder in which a user will enter his/her name; this placeholder should be a PDF text frame, not a simple text frame.
2. Double click on created PDF Text Frame which will be placeholder for all other similar entries, then give it a unique name (eg. FN1 for this first "master" box containing First Name which will be entered manually, then FN2 for the second First Name PDF frame etc.) In the example attached, I used FN notation for "First Name", and SN for "Second Name". Of course, you can adapt this to your liking.
3. Now, for FN1 you don't have to do anything, but for FN2 ... FNn you'll have to enter a simple script which will make sure content of FN1 is displayed in FN2 ... FNn. To do so, double click on FN2 PDF text frame, go to "Calculate" tab, check "Custom calculation script" > "Edit..." and enter this code:
var a = this.getField("FN1").value;
this.getField("FN2").value = a;
This basically defines a variable, here named "a" as an example (var a), which will be content of FN1 PDF frame and which will be automatically filled in FN2 (thus "value = a" in the second line of the code). After you've entered the code, go to "File > Save and Exit".
That's all. Export the file to PDF and test it.