That is because Lightroom uses ProPhoto. There is no option to use sRGB or anything else. Only ProPhoto in Lightroom.
ProPhoto is the largest gamut colorspace, so as you are exporting and opening in photoshop it sounds as if photoshop is ASSIGNING the colorspace as sRGB instead of converting it.
You could easily solve this problem by changing your photoshop to work in ProPhoto, but the problems there are many: Most to none of the monitors on the market can display all of the ProPhoto gamut. You can't print with your professional printer or a lab in prophoto, so eventually you will have to convert it anyway.
This is where I ran into too much trouble with Lightroom for it to be worth the headache for me. You cannot make your images in sRGB look the same as in ProPhoto RGB. There is just LESS information in sRGB gamut and it can't look the same. You can't make a gallon of gas out of a quart. You MUST be aware when working in LR of the clipping of the pure colors that will occur when you convert it. In the export menu there is an option to convert to sRGB. Be aware of how it will convert and look. If you are creating incredibly vibrant photos with a lot of high end pure colors your photos may well get a washed out look when converting.
Which is why I gave up and just use Adobe Camera Raw. ACR is the exact same engine as LR for the raw editing. Bridge does all LR does in cataloging, just not as intuitively. Open bridge and view your images and you can do all the same stuff as LR, only you work in the colorspace you choose.
EDIT TO ADD: You'll want 8-bit if you are doing any editing and or saving to jpeg. 16-bit and higher cannot be pixel edited or saved as jpegs.
