Selected Answer
See revision note below.
There may be an easier and more elegant way to do this but this seems to work...
In the attached file, I've added a series of dates in row 1 for what I think are your year end dates (sorry the format is European- I struggle to see things using mm/dd/yy format!). There's also a new hidden column Q (which makes copying formulae easier for me)
In typical "months" cell R5 say, the formula is:
=ROUND(12/360*IF(AND($J5<=R$1, $P5>=P$1),IF(AND($J5>=P$1,$P5<=R$1),$P5-$J5,MIN(R$1,$P5)-MAX(P$1,$J5)),0),0)
The outer ROUND(12/360....,0) bit converts and rounds a number of the days into months (I used 360 since the enddate in column P use EODATE which is on a 360 day basis I believe).
Inside that is IF(AND($J5<=R$1, $P5>=P$1),....,0) which checks the conditions when the project can have days worked in a given financial year (comparing project start and end dates with same for the current finacal year).
Within that is IF(AND($J5>=P$1,$P5<=R$1),$P5-$J5,MIN(R$1,$P5)-MAX(P$1,$J5)) where the AND checks if the project starts and ends in that FY and the true / false statements following that calculate the appropriate number of days.
Hope this helps.
REVISION
Apologies but my solution above added financial year end dates of November 30 (which I thought was very unusual) since the problem referred to the first project having 10 months on FY 2021 and 10 in FY 2021. In the file, the first project is 20 months starting Feb 1 however (which is 11 not 10 full months to the year end and 9 months in the subsequent year)
Also it was midnight local time when I posted that and I hadn't checked the hidden columns. Luckily @Variatus is very thorough and his answer prompted my to check. I saw you have start and end dates hidden in rows 2 and 3 respectively so I could:
1) modify the formula to refer to those instead, so new Q5 (I deleted the column I'd added) reads:
=ROUND(12/360*IF(AND($J5<=R$3, $P5>=R$2),IF(AND($J5>=R$2,$P5<=R$3),$P5-$J5,MIN(R$3,$P5)-MAX(R$2,$J5)),0),0)
2) removed the false dates I'd added in row 1.
In the attached revised file, I've also made the correction suggested by @Variatus for column P dates.
Please check this file (...v0_b.xlsm) is correct, noting that e.g. the first project "Chicago Project #1" in row 5 now shows 11 months in 2021 and 9 in 2022( not 10 + 10) .
Incidentally I could see no macros in your file but I didn't save it as the safer .xlsx in case you'd removed the macros before posting)