For clarity for anyone coming upon this later: what you've written is technically accurate but may not be evident to any of the people involved in a marriage (except the officiant).
What happens most of the time is that the government routinely licenses religious officials (priests, pastors, rabbis, imams, etc) to carry out the civil marriage. The religious marriage takes place, and the civil (government) parts of the marriage requirements are done in parallel (mainly signing marriage docs, civil registry documentation and the witnessing). The officiant of the wedding is acting as an agent of the government in this specific and limited sense - i.e. carries out the civil marriage as well as the religious one.
The easiest way to think of it is that (for the most part) religious marriages are two marriages in one (as long as the officiant is licensed). Legally though, only the civil part of the marriage counts. (Note also some variation by province)
I know this is a bit pedantic but it's precisely where this case causes issues in Canada. In some countries (much of Europe) the religious ceremony is legally meaningless or close to it, and only the civil marriage matters. In some other countries, the religious ceremony has the force of law, and civil registration (if it exists) may only be a legal formality or convenience.
I gather India is more like the latter case. Whereas someone who got 'married' in a church in France (I think) is not married in the eyes of the law in France (or therefore in Canada). (Note: saying this with no regard for common law union concepts or lots of other specifics or exceptions...)
I'm not really clear what your issue is or to whom you think you need to show or demonstrate your marital status. Obviously there are potential circumstances where it would matter, but a lot of cases where it won't matter or whichever organisation or entity you're dealing with simply won't ask for any documentation - they'll just take your word for it that you are spouses.
Also, if you've been living together for more than a year, then in most provinces, you are common law spouses (subject to some qualifications). The specifics and implications can vary a bit and have some peculiarities (depends on circumstances), but may be enough for most purposes (another reason why some may take word for it that you're married).
BUT: it is a very good idea to have legal documentation confirming your marital status. Copies with family members or a lawyer or a safe deposit box or whatever. If they're from out of country, extra copies, notarised, translated, legalised, etc - as needed or applicable. It may not be needed often, but if it is needed, it could take months and months, or worse require permissions or signatures from people not available, powers of attorney in multiple countries, etc. (Nobody wants to have to prove their marital status when somebody's sick or incapacitated)
Getting this may mean getting a copy of the marriage certificate or extract from the civil registry in the country where married, translation/notarised copy, and possibly legalised (e.g. through apostille or equivalent). Details vary by country. It's not necessarily that difficult but 'it depends.'
I gather from the Indian case referred to it may mean getting the religious ceremony registered first - in India - in order to get a civil registry certificate but no idea. There may be other approaches or suggestions to achieve the same ends under Canadian law, like powers of attorney, but that's for proper lawyers.
(To be fair from a foreign perspective the Canadian situation can also seem weird - there's no national registry to speak of, and some provinces still have archaic stuff on the books like the publication of the banns in Ontario)
Sorry if this post is a bit in the weeds.
