It's not clear to me what "all month round" or "all week round" would mean.
"All year round" effectively means "continuously". You start at January and then go right through until December, and then you repeat the cycle by continuing into the following January. It is a continuous cycle - or a continuous round.
So, by analogy, "all month round" would be a cycle repeating monthly: you start at the 1 January through to the end of January, and then continue into February, and so on, until you get to 31 Dec, when you continue to 1 January, ad infinitum. So in fact, that would be the same as "all year round".
The same argument would apply to "all week round".
So, if you are talking of something that happens continuously throughout the year, then why use the smaller calendar units, such as month or week?
On the other hand, if you are talking about only certain weeks, months or seasons, you would say (for example) "throughout the month", or "throughout the summer season". This is not round because it does not encompass a full circle/cycle and go back to the beginning of the same week/month/season. So words such as "through", "throughout", "long", etc. are more appropriate for that situation.
The Free Dictionary has the following definition:
all year round
Fig. throughout all the seasons of the year; during the entire year.
The public swimming pool is enclosed so that it can be used all year round.
In the South they can grow flowers all year round.
Although not expressly stated, the clear implication is that this is referring not to a single year, but to things that continue all year round - every year.