Western birthday once reaches the date, the person will plus one to their age. For example, born 2 November 1985, the person turns 35 on 2 November 2020.

How about lunar age? Does it works the same? Meaning Chinese lunar birthday is   九月三日，on 九月三日庚子年， the person officially turns lunar 36 age?

Yes the same.

But 12 animal zodiac only change on  LiChun 立春 and LiChun 立春 can be before or after Lunar Chinese New Year. If you born before LiChun 立春 and this LiChun 立春 is after Lunar Chinese New Year, you take last year animal sign. For example for 1979, Lunar New Chinese New Year is on 28 Jan, LiChun is on 4 Feb,  born on 29 Jan 1979 even it is 1月初二 but year pillar is 戊午 (1978 solar) and not 己未.(1979 solar) You are 1979, 1月初二 but you cannot say 1979 己未, 1月初二.  戊午 is 12:14pm  from 4 Feb 1978 to 0:13 am 4 Feb 1979

Cheers.