Here’s the bad news upfront: it is unlikely that you will be able to start a PhD in that particular lab, at least before September.
By far, the most common way of funding a PhD in France is that the lab secures funding before the student is found. If the professor thinks it is not possible, they probably are right. EDIT: this might be domain-dependent. That is the case at least in physics, see comments.
I believe there is no specific funding for Indian students in mathematics. That belief is based from my quick search on that website, but I have not read carefully all the hits for the relevant filters (Indian nationality, PhD level, mathematics). Campus France is the PR agency / embassy of FR higher ed for foreign students, so their website is probably the best place to start any search.
The rest of the answer is more about generalities for the way PhDs are funded in France.
In STEM at least, École Doctorales (aka ED, the administrative structures in charge of PhD students) require a proof of funding for three years before they register a PhD student. The exact minimum level of funding required varies across EDs, but it is typically around €1000/month - for ED184, it is €800/month as of February 2022.
Therefore, grants that cover only one year of study, or only some minimal costs, are insufficient, unless you can stack enough of them.
You need to decide at which point of time you would accept to start your PhD. Most PhD positions are filled between September and December (September is in sync with the school calendar of finishing master’s students, but having interviews and finding the good candidate etc. can delay a bit). Asking in December usually means most of the spots are already taken.
If you are OK with a start in September 2022, and you did not mention that in your initial email, you should probably follow-up on your email with that information, asking the professor if they expect some funding to pop up by then. (Even if they answer positively, the usual job-search advice applies - keep looking for positions until you actually have one, do not fixate on getting a position in this particular lab; certainly do not waste six months of job-hunting just waiting to see if that position gets funded.)
The "local situation" part seems intentionally vague, so I would not press on what that situation exactly is.
Funding: the most common scheme
EDIT: a large part of this section comes from my experience, which may not be representative of all STEM fields. In particular, the assertion that >90% of position open before the candidate is sought is probably incorrect in non-physics fields.
Usually, the lab secures funding prior to the PhD student arrival. This funding can come from a plethora of sources, but how strong you are as a candidate has zero influence over whether the lab can get them.
In contrast, there are many funding sources that are reserved to specific applicants, or require a joint application from the lab and prospective student. Those are rare, and usually very specific (students from some particular schools, military personnel, CNRS technician staff etc.). I would say those are <10% of PhD positions, probably <5%, though I could not find actual figures.
Of the pre-funded positions, some are filled from the master’s student pool from internship/classes, the rest are advertised and filled similarly to a non-academia job. I would think most of the positions that are advertised on the internet are actually open, because there is no requirement (that I know of) to publicize those (OTOH, tenured jobs must be publicized due to "open competition" rules, but will often be pre-filled by a local candidate, in which case applying is a waste of time). The job posting will have a fairly detailed outline of the research program (bibliographies and figures are not uncommon).
Unfortunately, I am not aware of an exhaustive central repository of open PhD positions (or open positions in math). Campus France has some, but it does not contain all positions. An exhaustive search requires to look for those on the university/lab websites, one by one - for instance, here are the postings at the place I did mine.
I am not sure every university website has English postings, and even among those that do, I would not trust every English translation to be correct, so I would advise to browse in French if you can and/or retranslate via machine tools the French original of whatever English posting seems interesting to you.
Impact of funding source on your day-to-day duties
I would not care much about the funding source while searching, but you may want to ask questions before accepting an offer. A few generalities from my limited experience:
- university/ED contracts usually mean significant teaching duties (might be a positive or a negative depending on how you see it); if you need a couple more months to finish the PhD beyond the official 3 years there is a decent chance to extend the contract in some fashion
- grant contracts (ERC or ANR [agence nationale de la recherche]) usually mean less or no teaching, but (from what I heard) are more red-tape-heavy
- CIFRE contracts (university/industry collab) usually mean better salary (compared to university contracts) and little to no teaching duties, but often entail rigid adherence to the research program (at the university, your professor cares that you publish good-quality science, but they will not care much if the topic is entirely different from what they thought it would be).