Different shapes of fluorescence spectra in Raman
There is no reason why fluorescence emission background should look similar at all with green and red excitation.

Different substances


You say the green excitation spectrum is of Coke. 
According to the video, the red excitation spectrum is of the peel of a tangerine.


Those samples consist of totally different substances so they will have different fluorescence spectra.

Different emission wavelength ranges

Assume this black line is the fluorescence emission spectrum of one sample:




the 532 nm excitation (dark green line) Raman spectrum is in the wavelength range 532 to 781 nm (green shaded area)
the 785 nm excitation (red line) Raman spectrum is in the wavelength range 801 to 1014 nm (red shaded area)


There isn't even any overlap between the spectral ranges you look at.

Different Excitation wavelengths

In first approximation (slight) changes in the excitation wavelength won't change the shape of the fluorescence emission, only the intensity.
However, if the excitiation wavelength changes sufficiently to excite a different fluorophore, also the shape of the fluorescence emission spectrum can (will) change.



About buying a Raman system

Before going deeper into the buying decision I suspect you'll need to learn more about the basics of Raman spectroscopy (and maybe about fluorescence). I recommend starting maybe with 
some textbook chapters about spectroscopy for chemical analysis in general e.g. Skoog et al.: Principles of Instrumental Analysis if you directly like to have a focus on instrumentation (possibly also get some introductory text about spectroscopy for structure determination), and then to get deeper into Raman McCreery: Raman Spectroscopy for Chemical Analysis, ISBN: 978-0-471-23187-5. 

Any serious advise about which system is better for you will need a whole lot of further background information: e.g.
what samples will you measure (phase, analytes, matrices), what applications (qualitative, quantitative, spatially resolved, restrictions on measurement time, lab use vs. field use, need for polarized measurements/depolarization ratio, ...),
will these specification change in future (other research projects).

As you comment that there is no Raman instrument in your country,


maybe you could visit a reputable lab that does Raman on similar systems as the one you want to study?
Raman system manufacturers are usually quite willing to demonstrate the ability of their instrument to measure what you need with a few of your samples, either by you sending them a couple of samples or even by you visiting them with your samples so you can try out the handling of the instrument yourself.
Do this at least with the few instuments are in your shortlist so you do have a basis for comparing actual performance for your samples.




A few comments about the Raman experiments in the linked youtube-video


Vodka-in-bottle spectrum: measuring through or on glass usually doesn't disturbn Raman spectra for green (e.g. 532 or 514 nm excitation) and also 633 nm red excitation is OK. With 785 nm excitation you'll get very bad background.
The reason why the handheld measurement works is its long working distance: the focus is deep inside the bottle, thus hardly any scattered light from the bottle gets to the detector. 
Sugar Raman measurement background: not all background in Raman spectroscopy is fluorescence. White powders (that are actually clear substances with lots of refraction and reflexion due to the small particle size) will give a lot of elastically scattered light (i.e. still excitation wavelength). Depending on how good the spectrograph is, such scattered light may end up in the detector at the wrong pixel and thus be interpreted to have another wavelength.
Such scattering background will usually decrease the further you get from the Rayleigh line. A conclusive experiment to show that it is scattering and not fluorescence is: dissolve (or melt) the powder and deposit the substance as a clear film (after drying). Scattering will then (mostly) be gone. 
Sugar vs. Sugar in fruit and the all-same-food spectra: solid sugar vs. sugar dissolved in water won't have the same vibrational spectrum (the spectra of some substances do not change much when dissolving in certain solvents, but sugar has tons of interaction with the surrounding water molecules due to its -OH groups - and that does affect the spectrum.)
As for the "almost all food has the same NIR spectrum": right in that wavelength range of 940 - 99 nm are overtones of -OH (water, carbohydrates, alcohols, H-bridges to oxygen). As all those fruits and veggies (including the peel) are in first approximation a mix of water and carbohydrate, there shouldn't be any surprise in those spectra being very similar.
With a good spectrometer and good data analysis (which includes good design of the calibration experiments) you can approach questions like "is this apple ripe for harvesting yet", but this requires the spectra to be available with a noise level (reproducibility) that allows measuring small shoulders of the absorption band.
Besides what the video discusses, there are a some other important things Tellspec claimed but cannot measure for physico-chemical reasons: mono-atomic ions such as Na⁺. They don't have molecular vibrations (again with super-good spectroscopic equipement you may be able to measure the effect of the ion on the water, but that's not going to work in practice). Which means, if you scan a soup and then add salt, it won't see the difference. Yet they claim to tell you sodium intake...