Should I consider replacing bottom bracket? I have a Cube Acid MTB from 2002. It has been used for 45000km, but only during summer on the road, and never in the rain, so absolutely no water on any parts.

Since I will be replacing other parts during the winter break by myself, I was planning on having the LBS to remove the BB so I could find a similar one, and then have them install it before next season.

But I am not sure it is needed. I was just thinking that 45000km sounded like a lot for a BB that has not been serviced for 16 years. The rides the last few months has felt a little bit "muddy", like when you walk in water there is some resistance, but that could be the chain which is a little bit dirty and old (Shimano HG93 XT 27000km).

I have just removed the chain, and when I pull the pedals I do not see any play and when I spin them, I do hear a little bit of noise, but it is more "whoosh" than any creaking or grinding noise, and it does seem to run smoothly.

I do not know exactly what BB is installed but it have to be some standard Shimano. So should I consider replacing it after 45000km in dry conditions or is it possible it can be used for many more miles?

Does anybody have a guess about what BB is installed? I can measure the axle length and diameter, but all this with square tapers and thread, I do not know anything about.
I would have though that after  18 years and 45000km the bottom bracket would be worn out. I recently learned that bearing wear is heavily dependent on load (source: GCN Everything You Need to Know About Bearings) so if you are a light rider and are not putting out lots of power I guess the bottom bracket bearings could last that long.

If you are not feeling any play in the bottom bracket bearings and they turn smoothly that's further evidence the bearings are OK. If you can you should pull the cranks off and turn the axle with your fingers. You may find you feel drag or 'notchiness' that the mass of the cranks is disguising.  

You very likely have a cartridge type bottom bracket bearing unit, these are not particularly expensive and are available from a number of vendors so you may as well replace it. I'd check the chain with a wear gauge and consider replacing it, and look at the chainrings and sprockets as well.