You can indeed, we pay £25 a month into the Scottish Mortgage IT via a bare trust for our son:

https://www.bailliegifford.com/individual-investors/how-to-invest/childrens-savings-plan/

We also do another £25 per month into the Blackrock Consensus 85 through a HL JISA:

http://www.hl.co.uk/investment-services/junior-isa

Yes but you wouldn't buy shares you'd buy funds or investment trusts.
Assuming this is an 18 year or so investment then statistically you would get much more from these investments than savings so all into investments rather than 50/50 woukd be the best thing to do,

I know Hargreaves Lansdown have a scheme that allows £50/month savings as that's what I do for my granddaughter. I'm sure others also allow regular payments at this level. It's all done by DD so once set up nothing to do.

Your other decision is whether to save specifically into your daughters name or do it in yours, earmarked for her. The benefit of the latter is that you aren't obliged to give it to her at 18. If she's a massive spendthrift for example you might decide to hold it back a few years. You can't do that if it's in her name it's hers to spend on whatever she wants when she is 18.

BTW these investments should be in a tax free account, which means either an ISA or a JISA (junior ISA). The disadvantage of it being in an ISA in your name is that her inv stments gets mixed in with yours.

This is what I do though and just have her fund that I invest in one that I don't buy myself so it's clear which part of the ISA is hers. E.g. Every month I put £50 into Fund A. I also have funds B,C,D. It may be I might sell those and buy E,F G instead or keep those amd also buy fund X. But I keep on buying A every month so it's clear what is hers, when it's time, she is due that value of all the A.

Hopefully this is clear if not ask .