Can We estimate the order of magnitude of number of mammals on earth? Can this be treated as Fermi problem or not?
Yes and no. Yes in the sense that I believe a Fermi-type scrutiny of the problem quickly tells you that you need very detailed data.
What may be derivable from such an analysis is an estimate of the total mammal biomass. A kilogram of living mammal has roughly the same energy needs whatever mammal it belongs to, although there would need to be a "fudge factor" to account for the animal's size: small animals tend to have very high surface area to mass ratios and thus high metabolic rates. You could hazard a guess Looking at the kinds of numbers of livestock farmers can successfully sustain per unit area in various parts of the world and you might hazard a guess as to what size of mammal mass there could be in the various climates.
But - and here is where you would be forced to shift from a Fermi type analysis to detailed research to get anything sensible - you would need to estimate what proportion of animals of each size makes up the mammal biomass. Is eight tonnes of mammal biomass one African Bull Elephant, or 160 000 mice at 50 grams each? Here I think we strike a problem: I can't see any way of "estimating" the size distribution, without detailed knowledge of population dynamics in each ecology.
Also of relevance may be the Biology Stack Exchange Question "How Many Mice Are There On The Earth?".