Can I register an unknown tablet?

I bought a tablet in Suning electronic chain store in Beijing 2 years ago, it cost 999 yuan, about £100 or €130. I can install some apps on it, but I don't seem to be able to do it directly from Google Play. I have to try to download the apk file & then use the apk installer. But this doesn't always work. So I'd love to know is there any way I can register my device with Google Play. It gives many tablet options, but not mine. Maybe it's a political issue, I don't know.
It's called efun M900, also called MID. (when I connect it says "You have connected your MID to your computer via USB...") It takes mini sd, also has NAND flash 2.81GB, internal storage 248MB. Last time I was on Google Play the message said "no eligible devices for app install".
"About Device" says
Model number M900
Android Version 2.3.1
Kernel version 2.6.32.27
Build number V1.0.17.GINGERBREAD.eng.root.20111014.005339_HW1.0.0.0

I also have a Sony Ericsson mini xperia, (ST15i) now running Android 4.0.4; Google Play does recognise this device.

Thank you

Kris, thanks for your input. I checked the video but have 2 issues before I proceed.
1. The link to the web site given in the text in 3a by RolandDeschain79 is different to the one shown in the video.
Clicking the link brings me to androidfilehost.com/?fid=23060877489998215
But I don't think this is the correct one.
The link shown on the video at 1m04s is 2shared.com/file/Z5DUeYSG/Play_Store_31014.html

I left a comment on youtube about this (posting as RichXZ)
I had to change references to websites in youtube comments. I don't know if there's an accepted way to do this, I just wrote "dot" for "." & youtube accepted it.
2. When I tried this the other day I was asked to register, & include credit card details, which I was reluctant to give, even though the site claimed they would not take any money unless I purchased something from them.

Thanks

Ok, Kris, thanks. here's how things stand at present

The first website androidfilehost.com/?fid=23060877489998215 invites me to download

com.android.vending-4.3.10.apk This may be another name for Google Play store (I don't know), but I didn't download it.

The 2shared.com/file/Z5DUeYSG/Play_Store_31014.html invites me to download Play Store

3.10.14.apk , which looks right

But when I click (using my tablet) it either offers a download of TorchSetup.pkg or invites me to sign

up with 2shared.com (depending which download button I click).

Then I have to register with email address & password, (I did this) which then brings me to a page

where I can upload files, but I can't see a way to download anything!

The FAQ said go to My Files, but I tried that & couldn't see anything. So instead I went to
2shared.com/file/Z5DUeYSG/Play_Store_31014.html on my PC, & successfully (surprise)
managed to download Play Store 3.10.14.apk, then copied it to the tablet.

This seemed to work. I now have an app on my tablet home page (is it called the
desktop?) called Play Store, with the same icon my phone has for Play Store. But when
I open the app, it says "You must add an account to the device to continue. Do you want
to add one now?" I click "Yes", but then the app just switches off. ("No" produces the same
result). I do have an email account with Google registered on the tablet. I switched off & on
again, but it's still the same.

So I'm not really sure if that's progress, but it did take time!

Richard