How long does it actually take, if the 180l tank has been emptied halfway, until the heat pump has "refilled" the 90l? Does this depend on the settings of the heat pump / photovoltaic power etc., or does the heat pump generally always start immediately to heat new cold water and make it available again in the tank?
A heat pump with 6kW heating capacity will heat 90l of water from 10 degrees to 45 degrees in 40 minutes. If your heat pump is stronger, e.g. 10kW, it will be correspondingly faster. If the heat pump is allowed to additionally use a possibly existing heating element (e.g. 6kW), it will be even faster. However, the heating element is of course inefficient. Normally, this is not done.
The second question is when a heat pump (or generally a heating system) starts producing hot water. Usually, this depends on the hysteresis, i.e. falling below a preset temperature in the tank. If the temperature falls below the limit, hot water production begins until reaching an upper threshold. For example, if you set 45 degrees hot water temperature with a hysteresis of 2 degrees, hot water production starts below 43 degrees and works up to the 47 degrees target temperature. Additionally, other values for the start/end of hot water production can also be configured. Joedreck mentioned the option to set blocking times during which no hot water may be produced. I personally find this somewhat inflexible, but if it fits the lifestyle, it can certainly be done. With constant shower/bath behavior this might work, with a family with shift work maybe less so. You yourself also mention the influence of photovoltaics. That can also be done, e.g. via the "SG ready" function of the heat pump. If the inverter of the photovoltaic system has an output to indicate solar power production, this can be coupled with such a heat pump and e.g. overload the hot water tank. Instead of 43/45/47 degrees one can e.g. overload by +5K because cheaper photovoltaic power is available in the house at the moment.
In the case of hot water tanks: with what water does the heat exchanger charge the hot water tank? The water heated by the brine-water heat pump to about 30 °C (supply temperature of the underfloor heating) is not sufficient there, is it? Is there an additional heating element or can the brine-water heat pump also do this with the compressor?
Hot water is a separate circuit. This runs normally via the compressor, but is less efficient due to the higher target temperature. Therefore, it is also tried to keep the hot water temperature as low as possible. The heating technician will happily set you >50 degrees storage temperature. This is nice because the hot water supply then lasts correspondingly longer, but is inefficient. 45 degrees is a good starting value. If you then find that the storage volume is too small for your own hot water consumption, you can turn it up and thus "buy" yourself more comfort at the expense of efficiency.