Pi is an irrational number. It can't be compared with quantities like populations. When writing irrational numbers with digits (in a context like this), there has to be some sort of indication that it's an approximation.  "pi (~3.14)" or "pi (approx. 3.14) would work but "pi (3.14)" is wrong.
I think "noun" is correct. 円周率 is a name like "golden ratio or "Euler's number". Oxford, Collins and Cambridge have a "number" PoS but they all classify "pi" as a noun.

I disagree, when I google "Tokyo population" I get an info box that reads "Tokyo population: 13.96 million". The 
official Tokyo metropolitan government site instead gives a number rounded up to 3 decimal places. Neither uses an 
ellipsis to indicate that the actual number (expressed in millions) is not actually 13.960000 or 13.961000. I notice 
most English dictionaries define pi like "approximately 3.14xxx" or something similar and that one Oxford Languages 
dictionary does use the ellipsis so I'm OK with showing it like this, for clarity.
I'm not sure whether number or noun is correct. Do most dictionaries have a [number] category to begin with? MW online 
has "2" as a pronoun/adjective, "3" as a noun.

"3.14" without an ellipsis would be incorrect. Also, I think it looks better with a few more significant figures.
"pi" is a noun in English dictionaries. And it shouldn't be capitalised.

@@ -11 +11 @@
-<pos>&num;</pos>
+<pos>&n;</pos>
@@ -14 +14 @@
-<gloss>Pi (3.1415926...)</gloss>
+<gloss>pi (3.1415926...)</gloss>

Rounded up to two decimal places, it is.
Is it weird having a num tagged as [math]?

@@ -11,0 +12 @@
+<field>&math;</field>

Pi is not 3.14.  I think what was there before was fine.

@@ -13 +13 @@
-<gloss>Pi (3.14)</gloss>
+<gloss>Pi (3.1415926...)</gloss>