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The monarch of the United Kingdom is the head of state of the sixteen Commonwealth realms. The question "What governmental power does the Monarch hold in Great Britain?" already explains the role of the British monarch in the United Kingdom.
What political significance does the monarch have in the other Commonwealth realms?
In Canada
I'd like to say "none" -- no political significance -- but don't know how to prove that.
That's been especially true, in Canada, since 1982.
There is a Governor General who has approximately the same role (quasi-ceremonial) in Canada as the Queen does in England ... or even less so, as the Governor General is a temporary position (whereas the Queen is permanent and may have more "soft power", and social/personal contacts):
The incumbent will generally serve for at least five years, though this is only a developed convention, and the governor general still technically acts at Her Majesty's pleasure (or the Royal Pleasure). The prime minister may therefore recommend to the Queen that the viceroy remain in her service for a longer period of time, sometimes upwards of more than seven years.
The Governor General isn't chosen by the queen (but presumably communicates with and/or visits to meet the Queen sometime[s]).
Very occasionally the Governor's General's decision, of whether to dissolve parliament, is non-trivial.
The Canadian Armed Forces still swear their allegiance to the Queen. I once fondly imagined that was a check-and-balance against a prime minister's using the armed forces against the people, but maybe that was naive (and not much of an example of the Queen's political power in Canada). On the other hand, something like this sentiment seems to be written on the Queen's own web site:
The Queen personifies the state and is the personal symbol of allegiance, unity and authority for all Canadians. Legislators, ministers, public services and members of the military and police all swear allegiance to The Queen. It is for this reason that all new Canadian citizens swear allegiance to The Queen of Canada. Elections are called and laws are promulgated in The Queen's name.
And of course the Queen is seen as the Head of State of the UK -- always a dignitary when she visits.
There are some royalists or monarchists in Canada but SFAIK they're politically insignificant (though I guess that, their being insignificant, is even more true of their counter-parts, the "republicans").
Victoria Day is a major holiday, not for especially royalist reasons ("informally considered the beginning of the summer season in Canada").
As for signing treaties, that's detailed here: Negotiation and Conclusion of International Treaties -- Canadian Practice:
It is widely held that the head of state is the only person capable of representing Canada internationally and the only person who has the power to sign international conventions or treaties on its behalf. The reality is quite different, however. Although the Governor in Council (Cabinet) retains final effective control over the ratification of treaties, it may appoint any person it wishes to negotiate and sign them. Apart from the Prime Minister, these persons are usually the ministers, deputy ministers, diplomatic representatives or negotiators of the Canadian government. As soon as the Governor in Council approves a treaty entered into by Canada and a foreign country, regardless of who has negotiated it, it becomes an international treaty, provided it is also ratified by the other signatory countries.
A treaty entered into and signed for and on behalf of Canada by a representative of the Government of Canada and subsequently approved by the Governor in Council is binding on Canada. Approval usually takes the form of an order in council.
I think that "the Governor in Council" is also known as "the Cabinet" or "the Privy Council", by the way, i.e. it's the Prime Minister and the Prime Minister's ministers ... it's not "the Governor General".
I'm not sure what role if any the Governor General plays in that. IMO the Governor General (like the Queen) never refuses to sign off on legislation enacted by the government (and isn't a member of the cabinet and doesn't attend cabinet meetings): and so, to the extent that the Governor General is involved, IMO it's only nominal/ceremonial/formal.
Order-in-council, at the federal level, is an order of the GOVERNOR GENERAL by and with the advice and consent of the Queen's PRIVY COUNCIL for Canada. In fact, it is formulated by CABINET or a committee of Cabinet and formally approved by the governor general.
Also FWIW it's not that she's the "British Monarch" that's significant ... rather it's that she's the Queen of Canada.
He or She also acts as head of state in those countries as well. Typically, this is through an appointment of a Governor-General who is effectively "keeping the local throne warm for them." The Governor-General is typically selected on the recommendation of the local Parliament. And keeping the throne warm is a bit of republicanism on my part, as they are appointed because they are in the know about the local politics a lot better than the Monarch, so they can make some minor emergency calls that would not require a months long journey by boat to deliver urgent news... or calling for help at 3 a.m. (The Sun never setting thing can be a pain on the royal beauty sleep).
In effect, the monarch is the symbolic figurehead of the state, and carries the same duties afforded in the United Kingdom, though there might be some differences (I know Elizabeth can't dissolve the Canadian Parliament like she can the UK Parliament).
