As others have said - this is a personal and subjective issue.
Also, you have not said what camera(s) you use and this affects the decisions.
I am arguably "semi professional" (or keen amateur) making money on my photography but not as my main job. Despite what Stan says, a very keen amateur does NEED a backup camera when it gets to he stage that taking photos in many situations becomes a significant part of the life experience. I have travelled extensively in Asia on business in recent years, I take many photos in "tourist mode" while away and I would not dream of not having a backup camera at all times. I'd be "lost" with no camera at all.
For worst case situations having a good enough pocketable of any sort is fine.
I carry a Sanyo XACTO video + still camera which literally goes in a (large) shirt pocket and which will produce 3/4/5/6 Mp stills (I have a number of them) and MPEG video (VGA through HD depending on model). Image quality is good enough for record-of-trip purposes. The main losses wrt a DSLR apart for sheer image size and quality are speed of composure, shutter response and frame rate. A good DSLR will capture a pigeon that suddenly flits low overhead, a child suddenly seen in mid jump, a sudden gesture or multiple frames of an interesting activity. A P&S usually will not do any or all of these well.
BUT for "serious stuff" another DSLR is needed.
The more it has in common with your "main " camera the better.
Body: Usually lenses at least are interchangeable across the whole range of a manufacturers cameras. You can carry a D800 and a D2 and have the lenses match. Or a 5DMkII and an EOS350. The main differences are full frame to APSC and the lack of ability of some newer bottom end bodies to focus drive some non SSM lenses. Apart from such issue, a very cheap body can often be carried that still has excellent photographic capability. A leading edge camera will have more mp, better focsing, faster frame rate, better low light operation and more bells and whistles BUT a bottom edge 3 year old DSLR will usually take extremely good photos in most conditions. Not so long ago the top Nikon wedding cameras boasted 3 megapixels.
Lens interchangeability is so easy to achieve that it would be unusual not to have it. The 2nd camera acts as a lens holder for the first where top results are needed. See above re SSm/ manual focus drive and FF versus APSC crop issues.
Battery - it is a really really good idea to be able to share batteries as you need fewer chargers and can easily carry spares for either camera.
Memory card - the ability to use the same memory cards is also a good idea but this is not as vital as batteries or lenses. If the main camera uses SDHC and the older one CFG or whatever, pocketing say 2 x CF of moderate capacity will usually suffice and they can often be bought while away (usually at outlandish pries). If needing different memory cards be sure to be able to download either (flexible card reader or have the USB cable with you). Don't forget the backup charger.
Flash Usually compatible across bodies.
What I do (fwiw):
I'm working my way up the Minolta-Sony chain with the hope that Sony will even now make a real camera that comes close to the D700.
Sensors for the Nikon D800 they make but themselves they, so far, cannot save :-(.
(A900/A850 are fine on resolution but cannot see in the dark).
I have A77, A700 (died), A200 (sold), 7D, 5D, ...
These are 24 / 12 / 10 / 6 / 6 megapixel.
I thought about buying a somewhat newer backup BUT the 6 mp 5D doesa very good job. For most purposes quality is fine, body size is small, lenses are the same as the A77. Alas batteries differ and memory is SDHC/CF. So I carry 5D body with my "second most likely to be wanted" lens on it. Overhead is batteries, some CF cards and a charger. Not a lot of weight or room. 5D is worth to close to $0 to be worth selling so as a backup it serves a useful role.