When buying back a put option, what happens to the buyer?

If I sell a put option and buy it later back, what does that mean for the buyer? Can he now sell the shares to my seller?

You are buying to close your short put. If your counter party is selling to close then the contract ceases to exist. If your counter party is selling to open then the contract just changes hands (he replaces you on the short side).

Also, what if I have a very deep ITM put option as a buyer, but I cannot sell it, because it is not liquid at this price?

You can always sell the deep ITM put because the market maker must provide a quote. However, his bid may be less than the intrinsic value of your put (a haircut).

You could try for some price improvement with your STC order but there is no incentive for the market maker or anyone else to give you the full intrinsic value. While waiting for a better fill, the price of the underlying could change and you could give back some of your intrinsic value.

To avoid this haircut, if you have the cash or appropriate margin, you could perform the same Discount Arbitrage that a market maker would do if you sold your put to him at a discount (buy stock, exercise put). Buy the stock first to avoid directional leg out risk.