Can I ask my adviser for a raise?

First, it is not quite a law yet as it still needs to go through conference and be signed by the president. Regardless, you can ask for a raise, but arguing for a raise on the basis that your personal situation has changed is not really a good strategy. Most people ask for a raise when they can make a case that they have been performing well.

Many advisors do not have spare money floating around. Stipends can be set at the department or university level. As I said in this answer graduate student tution is an odd thing. Universities may either automatically increase stipends or decrease tution. It is really too soon to know. Your best bet is to probably unionize and make sure the university begins treating students responsibly.

The issue, at least as I see it, is that advisors need to justify to funders why a PhD student is needed as opposed to a permanent RA or post doc. When PhD students are cheaper than the other alternatives, it is easy. In my field, the NIH post doc rate is about $40,000 a year. This means there is only a little gap in costs between a PhD student, after covering tuition and fees, and a post doc.

The problem is not really the change in tax code, but rather in the fact that full time researchers are being charged tuition. This is something, at least in my opinion, that needs to be addressed on a university/department level and not with the individual advisor.

Due to the recently passed tax bill by the senate, graduate students will incur more taxes on their income which is already small.

It is not law yet.

One of the professors in the department promised his students to compensate them by supplementary scholarships or awards.

They should not have done that. For something that affects literally every graduate student waiting for the university to figure out what they want to do before putting together some sort of patchwork solution is likely to be way more useful in the long term.

Can I ask my adviser to do the same thing to me?

You can certainly ask. But graduate student funding often comes from fixed pots of money, and lots of faculty don't just have piles of unallocated money sitting about. Further, many departments are aggressively opposed to having compensation in a department vary by advisor.

Beyond that, if you asked me, my answer would be "Lets wait and see what the university intends first."

Such a request is likely to go nowhere. Most departments have a "fixed" stipend system: people get a pre-determined stipend, although this can be augmented when needed. Usually, such "raises" are to "reward" students for bringing in outside money, or for moving from the master's phase of a program to the doctoral phase.

The better approach would be to organize with students across the university to see what can be done (perhaps lowering the amount of the tuition charge associated with the waiver?).